Journal articles: 'Implicit-Explicit schemes'

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Yang, Xiaozhong, and Lifei Wu. "An Efficient Parallel Approximate Algorithm for Solving Time Fractional Reaction-Diffusion Equations." Mathematical Problems in Engineering 2020 (August 26, 2020): 1–17. http://dx.doi.org/10.1155/2020/4524387.

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In this paper, we construct pure alternative segment explicit-implicit (PASE-I) and implicit-explicit (PASI-E) difference algorithms for time fractional reaction-diffusion equations (FRDEs). They are a kind of difference schemes with intrinsic parallelism and based on classical explicit scheme and classical implicit scheme combined with alternating segment technology. The existence and uniqueness analysis of solutions of the parallel difference schemes are given. Both the theoretical proof and the numerical experiment show that PASE-I and PASI-E schemes are unconditionally stable and convergent with second-order spatial accuracy and 2−α order time accuracy. Compared with implicit scheme and E-I (I-E) scheme, the computational efficiency of PASE-I and PASI-E schemes is greatly improved. PASE-I and PASI-E schemes have obvious parallel computing properties, which shows that the difference schemes with intrinsic parallelism in this paper are feasible to solve the time FRDEs.

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Handlovičová, Angela, and Karol Mikula. "Finite Volume Schemes for the Affine Morphological Scale Space (Amss) Model." Tatra Mountains Mathematical Publications 80, no. 3 (December 1, 2021): 53–70. http://dx.doi.org/10.2478/tmmp-2021-0031.

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Abstract Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter θ, 0 ≤ θ ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (θ = 0), semi-implicit, fully-implicit (θ = 1) and Crank-Nicolson (θ = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.

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Pan, Yueyue, Lifei Wu, and Xiaozhong Yang. "A New Class of Difference Methods with Intrinsic Parallelism for Burgers–Fisher Equation." Mathematical Problems in Engineering 2020 (August 14, 2020): 1–17. http://dx.doi.org/10.1155/2020/9162563.

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This paper proposes a new class of difference methods with intrinsic parallelism for solving the Burgers–Fisher equation. A new class of parallel difference schemes of pure alternating segment explicit-implicit (PASE-I) and pure alternating segment implicit-explicit (PASI-E) are constructed by taking simple classical explicit and implicit schemes, combined with the alternating segment technique. The existence, uniqueness, linear absolute stability, and convergence for the solutions of PASE-I and PASI-E schemes are well illustrated. Both theoretical analysis and numerical experiments show that PASE-I and PASI-E schemes are linearly absolute stable, with 2-order time accuracy and 2-order spatial accuracy. Compared with the implicit scheme and the Crank–Nicolson (C-N) scheme, the computational efficiency of the PASE-I (PASI-E) scheme is greatly improved. The PASE-I and PASI-E schemes have obvious parallel computing properties, which show that the difference methods with intrinsic parallelism in this paper are feasible to solve the Burgers–Fisher equation.

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Qin, Xiao, Xiaozhong Yang, and Peng Lyu. "A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation." AIMS Mathematics 6, no. 10 (2021): 11449–66. http://dx.doi.org/10.3934/math.2021663.

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<abstract><p>The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.</p></abstract>

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Wu, Lifei, and Xiaozhong Yang. "An Efficient Alternating Segment Parallel Difference Method for the Time Fractional Telegraph Equation." Advances in Mathematical Physics 2020 (March 2, 2020): 1–11. http://dx.doi.org/10.1155/2020/6897815.

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The fractional telegraph equation is a kind of important evolution equation, which has an important application in signal analysis such as transmission and propagation of electrical signals. However, it is difficult to obtain the corresponding analytical solution, so it is of great practical value to study the numerical solution. In this paper, the alternating segment pure explicit-implicit (PASE-I) and implicit-explicit (PASI-E) parallel difference schemes are constructed for time fractional telegraph equation. Based on the alternating segment technology, the PASE-I and PASI-E schemes are constructed of the classic explicit scheme and implicit scheme. It can be concluded that the schemes are unconditionally stable and convergent by theoretical analysis. The convergence order of the PASE-I and PASI-E methods is second order in spatial direction and 3-α order in temporal direction. The numerical results are in agreement with the theoretical analysis, which shows that the PASE-I and PASI-E schemes are superior to the classical implicit schemes in both accuracy and efficiency. This implies that the parallel difference schemes are efficient for solving the time fractional telegraph equation.

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Whitaker, Jeffrey S., and Sajal K. Kar. "Implicit–Explicit Runge–Kutta Methods for Fast–Slow Wave Problems." Monthly Weather Review 141, no. 10 (September 25, 2013): 3426–34. http://dx.doi.org/10.1175/mwr-d-13-00132.1.

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Abstract Linear multistage (Runge–Kutta) implicit–explicit (IMEX) time integration schemes for the time integration of fast-wave–slow-wave problems for which the fast wave has low amplitude and need not be accurately simulated are investigated. The authors focus on three-stage, second-order schemes and show that a scheme recently proposed by one of them (Kar) is unstable for purely oscillatory problems. The instability is reduced if the averaging inherent in the implicit part of the scheme is decentered, sacrificing second-order accuracy. Two alternative schemes are proposed with better stability properties for purely oscillatory problems. One of these utilizes a 3-cycle Lorenz scheme for the slow-wave terms and a trapezoidal scheme for the fast-wave terms. The other is a combination of two previously proposed schemes, which is stable for purely oscillatory problems for all fast-wave frequencies when the slow-wave frequency is less than a critical value. The alternative schemes are tested using a global spectral shallow-water model and a version of the NCEP operational global forecast model. The accuracy and stability of the alternative schemes are discussed, along with their computational efficiency.

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Yang, Xiao Zhong, and Gao Xin Zhou. "Improved Additive Operator Splitting Algorithms for Basket Option Pricing Model." Advanced Materials Research 756-759 (September 2013): 2739–43. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.2739.

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In order to solve Black-Scholes equation of basket option pricing model by numerical method. This paper used Additive Operator Splitting (AOS) algorithm to split the multi-dimensional Black-Scholes equation into equivalent one-dimensional equation set, and constructed 'Explicit-Implicit' and 'Implicit-Explicit' schemes to solve it. Then compatibility, stability and convergence of those schemes were analyzed. Finally, this paper compared computation time and precision of the schemes through numerical experiments. 'Explicit-Implicit' and 'Implicit-Explicit' schemes of AOS algorithms have both higher accuracy and faster computing speed and them have practical significance in solving basket option pricing model.

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Zhang, Shiyan, and Khalid Al-Asadi. "Evaluating the Effect of Numerical Schemes on Hydrological Simulations: HYMOD as A Case Study." Water 11, no. 2 (February 14, 2019): 329. http://dx.doi.org/10.3390/w11020329.

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The importance of numerical schemes in hydrological models has been increasingly recognized in the hydrological community. However, the relationship between model performance and the properties of numerical schemes remains unclear. In this study, we employed two types of numerical schemes (i.e., explicit Runge-Kutta schemes with different orders of accuracy and partially implicit Euler schemes with different implicit factors) in the hydrological model (HYMOD) to simulate the flow hydrograph of the Leaf River basin from 1948 to 1988. Results computed by different numerical schemes were compared and the relationships between model performance and two scheme properties (i.e., the order of accuracy and the implicit factor) were discussed. Results showed that the more explicit schemes generally lead to the overestimation of flow hydrographs, whereas the more implicit schemes lead to underestimation. In addition, the numerical error tended to decrease with increasing orders of accuracy. As a result, the optimal parameter sets found by low-order schemes significantly deviated from those found by the analytical solution. The findings of this study can provide useful implications for designing suitable numerical schemes for hydrological models.

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Hussain, Nawab, Vivek Kumar, Preety Malik, and Renu Chugh. "Jungck-type implicit iterative algorithms with numerical examples." Filomat 31, no. 8 (2017): 2303–20. http://dx.doi.org/10.2298/fil1708303h.

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We introduce a new Jungck-type implicit iterative scheme and study its strong convergence, stability under weak parametric restrictions in generalized convex metric spaces and data dependency in generalized hyperbolic spaces. We show thatnewintroduced iterative scheme has better convergence rate as compared to well known Jungck implicit Mann, Jungck implicit Ishikawa and Jungck implicit Noor iterative schemes. It is also shown that Jungck implicit iterative schemes converge faster than the corresponding Jungck explicit iterative schemes. Validity of our analytic proofs is shown through numerical examples. Our results are improvements and generalizations of some recent results of Khan et al.[21], Chugh et al.[8] and many others in fixed point theory.

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Durran, Dale R., and Peter N. Blossey. "Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems." Monthly Weather Review 140, no. 4 (April 2012): 1307–25. http://dx.doi.org/10.1175/mwr-d-11-00088.1.

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Implicit–explicit (IMEX) linear multistep methods are examined with respect to their suitability for the integration of fast-wave–slow-wave problems in which the fast wave has relatively low amplitude and need not be accurately simulated. The widely used combination of trapezoidal implicit and leapfrog explicit differencing is compared to schemes based on Adams methods or on backward differencing. Two new families of methods are proposed that have good stability properties in fast-wave–slow-wave problems: one family is based on Adams methods and the other on backward schemes. Here the focus is primarily on four specific schemes drawn from these two families: a pair of Adams methods and a pair of backward methods that are either (i) optimized for third-order accuracy in the explicit component of the full IMEX scheme, or (ii) employ particularly good schemes for the implicit component. These new schemes are superior, in many respects, to the linear multistep IMEX schemes currently in use. The behavior of these schemes is compared theoretically in the context of the simple oscillation equation and also for the linearized equations governing stratified compressible flow. Several schemes are also tested in fully nonlinear simulations of gravity waves generated by a localized source in a shear flow.

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Ruhiat, Yayat, and Suherman Suherman. "Development of Heat Conduction Equation using a Heat Propagation Model on ERK Solar Dryer Plates." Physics Access 04, no. 01 (May 2024): 44–50. http://dx.doi.org/10.47514/phyaccess.2024.4.1.005.

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The heat conduction equation is a combination of first-order and second-order differential equations. Solving first-order differential equations is necessary to examine temperature as a function of time. Meanwhile, solving second-order differential equations is needed to examine temperature as a function of space. The heat flux equation is based on Fourier's law, which shows that temperature is a function of time and space. Understanding heat conduction can be improved by building a heat propagation model on the Solar ERK dryer plate. Analysis of heat propagation on the drying plate used the Finite Difference Approach (FDA) method with explicit and implicit schemes. With an explicit scheme, the FDA method calculates the temperature (T) at a point on the spatial derivative term, when T is at time t, while the implicit scheme calculates T at a point on the space derivative term when T is at time t+Δt. Heat propagation at each time change was analyzed by developing a program using the MATLAB 17 application. The results of the analysis show that there are differences in heat propagation between the explicit and implicit schemes. The convergence and stability of calculations in explicit schemes are unstable, causing problems at the time step. Meanwhile, the implicit scheme is carried out simultaneously on all nodes so that convergence and stability are easily maintained, and there are no time-step limitations.

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Kim, Wooram, and J. N. Reddy. "A Comparative Study of Implicit and Explicit Composite Time Integration Schemes." International Journal of Structural Stability and Dynamics 20, no. 13 (August 8, 2020): 2041003. http://dx.doi.org/10.1142/s0219455420410035.

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In this paper, a number of recently proposed implicit and explicit composite time integration schemes are reviewed and critically evaluated. To give suitable guidelines of using them in practical transient analyses of structural problems, numerical performances of these schemes are compared through illustrative examples. Meaningful insights into computational aspects of the composite schemes are also provided. In the discussion, the role of the splitting ratio of the recent composite schemes is also investigated through a different point of view, and similarities and differences of various composite schemes are also studied. It is shown that the explicit composite scheme proposed recently by the authors can noticeably increase the efficiency and the accuracy of linear and nonlinear transient analyses when compared with other well-known composite schemes.

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13

Pieraccini, Sandra, and Gabriella Puppo. "Implicit–Explicit Schemes for BGK Kinetic Equations." Journal of Scientific Computing 32, no. 1 (January 23, 2007): 1–28. http://dx.doi.org/10.1007/s10915-006-9116-6.

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14

SUN, HONGGUANG, WEN CHEN, CHANGPIN LI, and YANGQUAN CHEN. "FINITE DIFFERENCE SCHEMES FOR VARIABLE-ORDER TIME FRACTIONAL DIFFUSION EQUATION." International Journal of Bifurcation and Chaos 22, no. 04 (April 2012): 1250085. http://dx.doi.org/10.1142/s021812741250085x.

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Variable-order fractional diffusion equation model is a recently developed and promising approach to characterize time-dependent or concentration-dependent anomalous diffusion, or diffusion process in inhomogeneous porous media. To further study the properties of variable-order time fractional subdiffusion equation models, the efficient numerical schemes are urgently needed. This paper investigates numerical schemes for variable-order time fractional diffusion equations in a finite domain. Three finite difference schemes including the explicit scheme, the implicit scheme and the Crank–Nicholson scheme are studied. Stability conditions for these three schemes are provided and proved via the Fourier method, rigorous convergence analysis is also performed. Two numerical examples are offered to verify the theoretical analysis of the above three schemes and illustrate the effectiveness of suggested schemes. The numerical results illustrate that, the implicit scheme and the Crank–Nicholson scheme can achieve high accuracy compared with the explicit scheme, and the Crank–Nicholson scheme claims highest accuracy in most situations. Moreover, some properties of variable-order time fractional diffusion equation model are also shown by numerical simulations.

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Wilson, Anastasia, Carson Morris, Kayli Hendricks, and Karen Lawrence. "A Comparison of First- and Second Order-in-Time Finite Difference Methods Applied to Nonlinear Reactive Transport." Mathematics Exchange 18, no. 1 (March 28, 2025): 2–25. https://doi.org/10.33043/28xy39zc.

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In this paper, we consider solution methods for the nonlinear reactive transport equation used to model the protein adsorption process. Efficient methods for simulating this process are necessary to aid in the development of novel adsorptive chromatography media to ensure high-volume production of purified product for the purposes of biotherapeutics. Using MATLAB®, we compare four finite difference schemes used to solve the nonlinear reactive transport equation, focusing on the differences of efficacy between implicit and explicit methods. As such, two of the methods are semi-implicit and two are explicit with one of each kind using a first-order temporal scheme and one of each using a second-order temporal scheme. The semi-implicit methods evaluate almost all terms implicitly while lagging the nonlinear coefficient function in time to linearize the equations. We include numerical results that indicate optimal convergence of the schemes, and we compare the effectiveness of the schemes in matching experimental data using two different boundary conditions.

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Huang, Jiancai, Shenghua Wang, and Yeol Je Cho. "Implicit and Explicit Iterations with Meir-Keeler-Type Contraction for a Finite Family of Nonexpansive Semigroups in Banach Spaces." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/720192.

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We introduce an implicit and explicit iterative schemes for a finite family of nonexpansive semigroups with the Meir-Keeler-type contraction in a Banach space. Then we prove the strong convergence for the implicit and explicit iterative schemes. Our results extend and improve some recent ones in literatures.

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Atangana, Abdon, and Dumitru Baleanu. "Numerical Solution of a Kind of Fractional Parabolic Equations via Two Difference Schemes." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/828764.

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A kind of parabolic equation was extended to the concept of fractional calculus. The resulting equation is, however, difficult to handle analytically. Therefore, we presented the numerical solution via the explicit and the implicit schemes. We presented together the stability and convergence of this time-fractional parabolic equation with two difference schemes. The explicit and the implicit schemes in this case are stable under some conditions.

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Ehigie, J. O., S. A. Okunuga, and A. B. Sofoluwe. "3-Point Block Methods for Direct Integration of General Second-Order Ordinary Differential Equations." Advances in Numerical Analysis 2011 (August 3, 2011): 1–14. http://dx.doi.org/10.1155/2011/513148.

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A Multistep collocation techniques is used in this paper to develop a 3-point explicit and implicit block methods, which are suitable for generating solutions of the general second-order ordinary differential equations of the form . The derivation of both explicit and implicit block schemes is given for the purpose of comparison of results. The Stability and Convergence of the individual methods of the block schemes are investigated, and the methods are found to be 0-stable with good region of absolute stability. The 3-point block schemes derived are tested on standard mechanical problems, and it is shown that the implicit block methods are superior to the explicit ones in terms of accuracy.

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Beubalayev, Vetlugin, and Abutrab Aliverdiev. "Numerical solution of the boundary value problem for the heat equation with fractional Riesz derivative." Thermal Science and Engineering 6, no. 2 (November 23, 2023): 2082. http://dx.doi.org/10.24294/tse.v6i2.2082.

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The work is devoted to the numerical solution of the initial boundary value problem for the heat equation with a fractional Riesz derivative. Explicit and implicit difference schemes are constructed that approximate the boundary value problem for the heat equation with a fractional Riesz derivative with respect to the coordinate. In the case of an explicit difference scheme, a condition is obtained for the time step at which the difference scheme converges. For an implicit difference scheme, a theorem on unconditional convergence is proved. An example of a numerical calculation using an implicit difference scheme is given. It has been established that when passing to a fractional derivative, the process of heat propagation slows down.

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Golubev, V. I., and I. S. Nikitin. "Refined Schemes for Computing the Dynamics of Elastoviscoplastic Media." Журнал вычислительной математики и математической физики 63, no. 10 (October 1, 2023): 1674–86. http://dx.doi.org/10.31857/s0044466923100046.

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For a stable numerical solution of the system of equations governing an elastoviscoplastic continuous medium model, a second-order explicit-implicit scheme is proposed. An explicit approximation is used for the equations of motion, and an implicit approximation, for the constitutive relations containing a small relaxation time parameter in the denominator of the nonlinear free terms. A second-order implicit approximation for isotropic and anisotropic elastoviscoplastic models is constructed to match the orders of approximation of the explicit elastic and implicit adjustment steps. Refined formulas for correcting the stress deviators after the elastic step are derived for various viscosity function representations. The resulting solutions of the second-order implicit approximation for the stress deviators of the elastoviscoplastic equations admit passage to the limit as the relaxation time tends to zero. The correcting formulas obtained via this passage to the limit can be treated as regularizers of the numerical solutions to the elastoplastic systems.

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Safwandi, Safwandi, Syamsul Rizal, and Tarmizi Tarmizi. "SEMI-IMPLICIT NUMERICAL SCHEMA IN SHALLOW WATER EQUATION." Jurnal Natural 17, no. 2 (September 6, 2017): 102. http://dx.doi.org/10.24815/jn.v17i2.7998.

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Abstract. A two-dimensional shallow water equation integrated on depth water based on finite differential methods. Numerical solutions with different methods consist of explicit, implicit and semi-implicit schemes. Different methods of shallow water equations expressed in numerical schemes. For bottom-friction is described in semi-implicitly. This scheme will be more flexible for initial values and boundary conditions when compared to the explicit schemes. Keywords: 2D numerical models, shallow water equations, explicit and semi-implicit schema.Reference Hassan, H. S., Ramadan, K. T., Hanna, S. N. 2010. Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method, Scie.Res,App.Math. (1):104-117. Omer, S, Kursat, K. 2011. High-Order Accurate Spectral Difference Method For Shallow Water Equations. IJRRAS6. Vol. 6. No. 1. Kampf, J. 2009. Ocean Modelling for Beginners. Springer Heidelberg Dordrecht. London, New York. Wang, Z. L., Geng, Y. F. 2013. Two-Dimensional Shallow Water Equations with Porosity and Their Numerical scheme on Unstructured Grids. J. Water Science and Engineering. Vol. 6, No. 1, 91-105. Saiduzzaman, Sobuj. 2013. Comparison of Numerical Schemes for Shallow Water Equation. Global J. of Sci. Fron. Res. Math. and Dec. Sci. Vol. 13 (4). Sari, C. I., Surbakti, H., Fauziyah., Pola Sebaran Salinatas dengan Model Numerik Dua Dimensi di Muara Sungai Musi. Maspari J. Vol. 5 (2): 104-110. Bunya, B., Westerink, J. J. dan Shinobu, Y. 2004. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids 2005 (47): 1451–1468.

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Yang, Xiaozhong, and Lifei Wu. "A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model." Mathematics 8, no. 4 (April 15, 2020): 596. http://dx.doi.org/10.3390/math8040596.

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Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

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Vabishchevich, P. N. "Explicit–Implicit Schemes for First-Order Evolution Equations." Differential Equations 56, no. 7 (July 2020): 882–89. http://dx.doi.org/10.1134/s0012266120070071.

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Vabishchevich, P. N., and M. V. Vasil’eva. "Explicit-implicit schemes for convection-diffusion-reaction problems." Numerical Analysis and Applications 5, no. 4 (October 2012): 297–306. http://dx.doi.org/10.1134/s1995423912040027.

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Briani, Maya, Roberto Natalini, and Giovanni Russo. "Implicit–explicit numerical schemes for jump–diffusion processes." Calcolo 44, no. 1 (March 2007): 33–57. http://dx.doi.org/10.1007/s10092-007-0128-x.

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Konor, Celal S., and Akio Arakawa. "Multipoint Explicit Differencing (MED) for Time Integrations of the Wave Equation." Monthly Weather Review 135, no. 11 (November 1, 2007): 3862–75. http://dx.doi.org/10.1175/2007mwr1923.1.

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Abstract For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

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Alam, Md Joni, Ahmed Ramady, M. S. Abbas, K. El-Rashidy, Md Tauhedul Azam, and M. Mamun Miah. "Numerical Investigation of the Wave Equation for the Convergence and Stability Analysis of Vibrating Strings." AppliedMath 5, no. 1 (February 19, 2025): 18. https://doi.org/10.3390/appliedmath5010018.

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The modeling of the one-dimensional wave equation is the fundamental model for characterizing the behavior of vibrating strings in different physical systems. In this work, we investigate numerical solutions for the one-dimensional wave equation employing both explicit and implicit finite difference schemes. To evaluate the correctness of our numerical schemes, we perform extensive error analysis looking at the L1 norm of error and relative error. We conduct thorough convergence tests as we refine the discretization resolutions to ensure that the solutions converge in the correct order of accuracy to the exact analytical solution. Using the von Neumann approach, the stability of the numerical schemes are carefully investigated so that both explicit and implicit schemes maintain the stability criteria over simulations. We test the accuracy of our numerical schemes and present a few examples. We compare the solution with the well-known spectral and finite element method. We also show theoretical proof of the stability and convergence of our numerical scheme.

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Caplan, Ronald M., Craig D. Johnston, Lars K. S. Daldoff, and Jon A. Linker. "Advancing parabolic operators in thermodynamic MHD models II: Evaluating a Practical Time Step Limit for Unconditionally Stable Methods." Journal of Physics: Conference Series 2742, no. 1 (April 1, 2024): 012020. http://dx.doi.org/10.1088/1742-6596/2742/1/012020.

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Abstract Unconditionally stable time stepping schemes are useful and often practically necessary for advancing parabolic operators in multi-scale systems. However, serious accuracy problems may emerge when taking time steps that far exceed the explicit stability limits. In our previous work, we compared the accuracy and performance of advancing parabolic operators in a thermodynamic MHD model using an implicit method and an explicit super time-stepping (STS) method. We found that while the STS method outperformed the implicit one with overall good results, it was not able to damp oscillatory behavior in the solution efficiently, hindering its practical use. In this follow-up work, we evaluate an easy-to-implement method for selecting a practical time step limit (PTL) for unconditionally stable schemes. This time step is used to ‘cycle’ the operator-split thermal conduction and viscosity parabolic operators. We test the new time step with both an implicit and STS scheme for accuracy, performance, and scaling. We find that, for our test cases here, the PTL dramatically improves the STS solution, matching or improving the solution of the original implicit scheme, while retaining most of its performance and scaling advantages. The PTL shows promise to allow more accurate use of unconditionally stable schemes for parabolic operators and reliable use of STS methods.

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YEE, H. C., and P. K. SWEBY. "GLOBAL ASYMPTOTIC BEHAVIOR OF ITERATIVE IMPLICIT SCHEMES." International Journal of Bifurcation and Chaos 04, no. 06 (December 1994): 1579–611. http://dx.doi.org/10.1142/s0218127494001210.

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The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2×2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

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Kazashi, Yoshihito, Fabio Nobile, and Eva Vidličková. "Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations." Numerische Mathematik 149, no. 4 (November 17, 2021): 973–1024. http://dx.doi.org/10.1007/s00211-021-01241-4.

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AbstractWe consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.

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FLOURI, EVANGELIA T., JOHN A. EKATERINARIS, and NIKOLAOS A. KAMPANIS. "HIGH-ORDER ACCURATE NUMERICAL SCHEMES FOR THE PARABOLIC EQUATION." Journal of Computational Acoustics 13, no. 04 (December 2005): 613–39. http://dx.doi.org/10.1142/s0218396x05002888.

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Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

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Zhang, Laiping, Ming Li, Wei Liu, and Xin He. "An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids." Communications in Computational Physics 17, no. 1 (December 19, 2014): 287–316. http://dx.doi.org/10.4208/cicp.091113.280714a.

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AbstractA Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was investigated also. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.

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Djumayozov, Umidjon, and Nigora Eshmanova. "Coupled Problem on Thermo-Elasticity in Strains for an Isotropic Parallelepiped." E3S Web of Conferences 497 (2024): 02016. http://dx.doi.org/10.1051/e3sconf/202449702016.

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This work is devoted to mathematical and numerical modeling of the coupled dynamic problem of thermoelasticity in deformations. A numerically related boundary value problem of thermoelasticity in deformations for a parallelepiped with the corresponding initial and boundary conditions is formulated and solved. Grid equations are constructed using the finite-difference method in the form of explicit and implicit schemes. In this case, the solution of the explicit scheme is reduced to recurrent relations with respect to deformations and temperature. In the case of implicit difference schemes, the equations are solved by sequential application of the sweep method. The validity of the formulated boundary value problems is justified by comparing the numerical results obtained with different methods based on two different models.

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Fernández, Miguel A. "Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit." SeMA Journal 55, no. 1 (September 2011): 59–108. http://dx.doi.org/10.1007/bf03322593.

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Shashkin, Vladimir V., and Gordey S. Goyman. "Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid." Russian Journal of Numerical Analysis and Mathematical Modelling 35, no. 6 (December 16, 2020): 355–66. http://dx.doi.org/10.1515/rnam-2020-0029.

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AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

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Gao, Jingyu, Maxim Smirnov, Maria Smirnova, and Gary Egbert. "A Comparison Study of Explicit and Implicit 3-D Transient Electromagnetic Forward Modeling Schemes on Multi-Resolution Grid." Geosciences 11, no. 6 (June 15, 2021): 257. http://dx.doi.org/10.3390/geosciences11060257.

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This study compares the efficiency of 3-D transient electromagnetic forward modeling schemes on the multi-resolution grid for various modeling scenarios. We developed time-domain finite-difference modeling based on the explicit scheme earlier. In this work, we additionally implement 3-D transient electromagnetic forward modeling using the backward Euler implicit scheme. The iterative solver is used for solving the system of equations and requires a proper initial guess that has significant effect on the convergence. The standard approach usually employs the solution of a previous time step as an initial guess, which might be too conservative. Instead, we test various initial guesses based on the linear extrapolation or linear combination of the solutions from several previous steps. We build up the implicit scheme forward modeling on the multi-resolution grid, which allows for the adjustment of the horizontal resolution with depth, hence improving the performance of the forward operator. Synthetic examples show the implicit scheme forward modeling using the linearly combined initial guess estimate on the multi-resolution grid additionally reduces the run time compared to the standard initial guess approach. The result of comparison between the implicit scheme developed here with the previously developed explicit scheme shows that the explicit scheme modeling is more efficient for more conductive background models often found in environmental studies. However, the implicit scheme modeling is more suitable for the simulation with highly resistive background models, usually occurring in mineral exploration scenarios. Thus, the inverse problem can be solved using more efficient forward solution depending on the modeling setup and background resistivity.

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Karaa, Samir. "Finite Element θ-Schemes for the Acoustic Wave Equation." Advances in Applied Mathematics and Mechanics 3, no. 1 (April 2011): 181–203. http://dx.doi.org/10.4208/aamm.10-m1018.

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AbstractIn this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1 + ∆ts), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and ∆t the time step.

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Khaldjigitov, Abduvali, Umidjon Djumayozov, and Dilnoza Sagdullaeva. "Numerical Solution of Coupled Thermo-Elastic-Plastic Dynamic Problems." Mathematical Modelling of Engineering Problems 8, no. 4 (August 31, 2021): 510–18. http://dx.doi.org/10.18280/mmep.080403.

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The article considers a numerical method for solving a two-dimensional coupled dynamic thermoplastic boundary value problem based on deformation theory of plasticity. Discrete equations are compiled by the finite-difference method in the form of explicit and implicit schemes. The solution of the explicit schemes is reduced to the recurrence relations regarding the components of displacement and temperature. Implicit schemes are efficiently solved using the elimination method for systems with a three diagonal matrix along the appropriate directions. In this case, the diagonal predominance of the transition matrices ensures the convergence of implicit difference schemes. The problem of a thermoplastic rectangle clamped from all sides under the action of an internal thermal field is solved numerically. The stress-strain state of a thermoplastic rectangle and the distribution of displacement and temperature over various sections and points in time have been investigated.

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Lock, S. J., N. Wood, and H. Weller. "Numerical analyses of Runge-Kutta implicit-explicit schemes for horizontally explicit, vertically implicit solutions of atmospheric models." Quarterly Journal of the Royal Meteorological Society 140, no. 682 (February 12, 2014): 1654–69. http://dx.doi.org/10.1002/qj.2246.

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Čiegis, Raimondas. "NUMERICAL SOLUTION OF HYPERBOLIC HEAT CONDUCTION EQUATION." Mathematical Modelling and Analysis 14, no. 1 (March 31, 2009): 11–24. http://dx.doi.org/10.3846/1392-6292.2009.14.11-24.

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Hyperbolic heat conduction problem is solved numerically. The explicit and implicit Euler schemes are constructed and investigated. It is shown that the implicit Euler scheme can be used to solve efficiently parabolic and hyperbolic heat conduction problems. This scheme is unconditionally stable for both problems. For many integration methods strong numerical oscillations are present, when the initial and boundary conditions are discontinuous for the hyperbolic problem. In order to regularize the implicit Euler scheme, a simple linear relation between time and space steps is proposed, which automatically introduces sufficient amount of numerical viscosity. Results of numerical experiments are presented.

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Mani Aouadi, S., W. Mbarki, and N. Zemzemi. "Stability Analysis of Decoupled Time-stepping Schemes for the Specialized Conduction System/myocardium Coupled Problem in Cardiology." Mathematical Modelling of Natural Phenomena 12, no. 5 (2017): 208–39. http://dx.doi.org/10.1051/mmnp/201712513.

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The Purkinje network is the rapid conduction system in the heart. It ensures the physiological spread of the electrical wave in the ventricles. In this work, we consider a problem that models the coupling between the Purkinje network and the myocardium. We first prove the stability of the space semi-discretized problem. Then we present four different strategies for solving the Purkinje/ myocardium coupling. The strategies are based on different time discretization of the coupling terms. The first scheme is fully coupled, where the coupling terms are considered implicit. The second and the third schemes are based on Gauss-Seidel time-splitting schemes where one coupling term is considered explicit and the other is implicit. The last is a Jacobi-like time-splitting scheme where both coupling terms are considered explicit. Our main result is the proof of the stability of the three considered schemes under the same restriction on the time step. Moreover, we show that the energy of the problem is slightly affected by the time-splitting schemes. We illustrate the theoretical result by different numerical simulations in 2D. We also conduct 3D simulations using physiologically detailed ionic models.

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Michel-Dansac, Victor, and Andrea Thomann. "TVD-MOOD schemes based on implicit-explicit time integration." Applied Mathematics and Computation 433 (November 2022): 127397. http://dx.doi.org/10.1016/j.amc.2022.127397.

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Ditkowski, Adi, Sigal Gottlieb, and Zachary J. Grant. "Explicit and implicit error inhibiting schemes with post-processing." Computers & Fluids 208 (August 2020): 104534. http://dx.doi.org/10.1016/j.compfluid.2020.104534.

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Pieraccini, Sandra, and Gabriella Puppo. "Microscopically implicit–macroscopically explicit schemes for the BGK equation." Journal of Computational Physics 231, no. 2 (January 2012): 299–327. http://dx.doi.org/10.1016/j.jcp.2011.08.027.

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Хамидуллин, М. Р., and А. Б. Мазо. "Accelerated explicit-implicit algorithms for the simulation of two-phase flow toward a horizontal multistage hydraulically fractured well." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 3 (August 31, 2017): 204–13. http://dx.doi.org/10.26089/nummet.v18r318.

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Представлены явно-неявные алгоритмы ускорения счета для решения трехмерной задачи двухфазной фильтрации вблизи горизонтальной скважины, пересеченной трещиной многостадийного гидроразрыва пласта. Ускорение достигается за счет ввода локальных зон, в каждой из которых применяется, в зависимости от локального числа Куранта, явная либо неявная схема для уравнения переноса насыщенности. Explicit-implicit approximation schemes for the numerical simulation of 3D two-phase flows toward a horizontal multistage hydraulically fractured well are proposed. This approach is based on the division of the computational domain into local subdomains and on using either an explicit or implicit scheme for the saturation transport equation with consideration of a local Courant number.

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POLIASHENKO, MAXIM, and CYRUS K. AIDUN. "COMPUTATIONAL DYNAMICS OF ORDINARY DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 05, no. 01 (February 1995): 159–74. http://dx.doi.org/10.1142/s0218127495000132.

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Discrete schemes, used to perform time integration of ODE’s, are expected to exhibit qualitatively ‘true’ dynamics in terms of the solutions and their stability. In past years, it has been discovered that such discretizations may cause spurious steady states and some explicit schemes may produce ‘computational chaos.’ In this study, we show that implicit time integration schemes, such as the backward Euler method, can also produce computationally chaotic solutions. Furthermore, we show that the opposite phenomenon may also take place both for explicit and for implicit schemes: computationally generated ‘spurious order’ may replace the true chaotic solution before the scheme becomes linearly unstable. The numerical solution may become chaotic again as the discretization step is further increased. The spurious computational order and chaos are discussed by solving low-dimensional dynamical systems, as well as a large system of ODE representing the solution to the Navier-Stokes equation. Our results support the point of view that the deviations in the behavior of the computed solution from the true solution has deterministic character with the time step playing the role of an artificial bifurcation parameter.

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Dehghan, Mehdi. "New schemes for a two-dimensional inverse problem with temperature overspecification." Mathematical Problems in Engineering 7, no. 3 (2001): 283–97. http://dx.doi.org/10.1155/s1024123x0100165x.

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Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and the (3,9) alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU) times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.

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Baysal, Oktay. "Supercomputing of Supersonic Flows Using Upwind Relaxation and MacCormack Schemes." Journal of Fluids Engineering 110, no. 1 (March 1, 1988): 62–68. http://dx.doi.org/10.1115/1.3243512.

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The impetus of this paper is the comparative applications of two numerical schemes for supersonic flows using computational algorithms tailored for a supercomputer. The mathematical model is the conservation form of Navier-Stokes equations with the effect of turbulence being modeled algebraically. The first scheme is an implicit, unfactored, upwind-biased, line-Gauss-Seidel relaxation scheme based on finite-volume discretization. The second scheme is the explicit-implicit MacCormack scheme based on finite-difference discretization. The best overall efficiences are obtained using the upwind relaxation scheme. The integrity of the solutions obtained for the example cases is shown by comparisons with experimental and other computational results.

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Lischke, Anna, James F. Kelly, and Mark M. Meerschaert. "Mass-conserving tempered fractional diffusion in a bounded interval." Fractional Calculus and Applied Analysis 22, no. 6 (December 18, 2019): 1561–95. http://dx.doi.org/10.1515/fca-2019-0081.

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Abstract Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. A reflecting boundary condition enforces mass conservation on a bounded interval. In this work, explicit and implicit Euler schemes for tempered fractional diffusion with discrete reflecting or absorbing boundary conditions are constructed. Discrete reflecting boundaries are formulated such that the Euler schemes conserve mass. Conditional stability of the explicit Euler methods and unconditional stability of the implicit Euler methods are established. Analytical steady-state solutions involving the Mittag-Leffler function are derived and shown to be consistent with late-time numerical solutions. Several numerical examples are presented to demonstrate the accuracy and usefulness of the proposed numerical schemes.

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Vidal-Salle, Emmanuelle, Charlotte Florimond, and Philippe Boisse. "Numerical Prediction of Internal Stresses due to Weaving." Key Engineering Materials 651-653 (July 2015): 338–43. http://dx.doi.org/10.4028/www.scientific.net/kem.651-653.338.

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The increasing use of finite element simulation in the field of composite material forming involved in the past few years a large amount of research on the constitutive modelling of textile material at the mesoscopic scale (i.e. the scale of individual fiber tow). Up to now, the community interest was focused on a consistent shape prediction. Moreover, the large amount of contacts between yarns imposed the use of dynamic explicit approaches for numerical efficiency reasons. Recent advances in contact algorithms make now possible the use of implicit schemes. The present paper shows how a constitutive equation written and implemented in the dynamic explicit scheme with ABAQUS/Explicit is adapted to implicit one (i.e. ABAQUS/Standard), for large displacement analyses.Validation and perspectives are illustrated on a weaving operation.

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Journal articles on the topic 'Implicit-Explicit schemes'

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