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Journal articles on the topic 'Upwind scheme'

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Published: 4 June 2021

Last updated: 19 June 2025

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Сухинов, А. И., А. Е. Чистяков, and Е. А. Проценко. "Upwind and standard leapfrog difference schemes." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 2 (March 28, 2019): 170–81. http://dx.doi.org/10.26089/nummet.v20r216.

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Для решения задачи переноса в статье предложено использовать схему, построенную на основе линейной комбинации разностной схемы кабаре (англ. Upwind Leapfrog) и крест (англ. Standard Leapfrog) с весовыми коэффициентами, полученными в результате минимизации погрешности аппроксимации. Проведено сравнение расчетов для задачи переноса на основе предложенной схемы с результатами, полученными с использованием схемы, построенной на основе линейной комбинации схемы с центральными разностями и схемы кабаре, и двухпараметрической разностной схемы третьего порядка точности. In order to solve the transfer problem, it is proposed to use the scheme based on a linear combination of the upwind and standard leapfrog difference schemes with weighting coefficients obtained by minimizing the approximation error. The estimate of the approximation error of the proposed difference scheme shows that, for small Courant numbers, this scheme whose approximation error is $O(ch^2)$, where the constant $c$ is significantly less than unity, is preferable to use than the original upwind and standard leapfrog schemes whose approximation errors are $O(h^2)$. The numerical results for the transfer problem based on the proposed scheme are compared with the results obtained using the following schemes: (i) the scheme based on a linear combination of the standard leapfrog scheme and the upwind leapfrog sscheme and (ii) the two-parameter difference scheme of the third order of accuracy.

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Bragin, M. D. "Upwind bicompact schemes for hyperbolic conservation laws." Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ 517, no. 1 (2024): 50–56. http://dx.doi.org/10.31857/s2686954324030097.

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For the first time, upwind bicompact schemes of third order approximation in space are presented. A formula is obtained for the transition factor of an arbitrary fully discrete bicompact scheme with integration in time by a Runge–Kutta method. Stability and monotonicity of the first-order in time scheme are investigated, dissipative and dispersion properties of the third-order in time scheme are analyzed. Advantages of the new schemes relative to their centered counterparts are demonstrated.

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Stynes, Martin, and Hans-Görg Roos. "The Midpoint Upwind Scheme." Irish Mathematical Society Bulletin 0038 (1997): 68. http://dx.doi.org/10.33232/bims.0038.68.

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Stynes, Martin, and Hans-Görg Roos. "The midpoint upwind scheme." Applied Numerical Mathematics 23, no. 3 (1997): 361–74. http://dx.doi.org/10.1016/s0168-9274(96)00071-2.

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Tkalich, Pavel. "Derivation of high-order advection–diffusion schemes." Journal of Hydroinformatics 8, no. 3 (2006): 149–64. http://dx.doi.org/10.2166/hydro.2006.008.

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Using the interpolation polynomial method, major upwind explicit advection–diffusion schemes of up to fifth-order accuracy are rederived and their properties are explored. The trend emerges that the higher the order of accuracy of an advection scheme, the easier is the task of scheme stabilization and wiggling suppression. Thus, for a certain range of the turbulent diffusion coefficient, the stability interval of third- and fifth-order up-upwind explicit schemes can be extended up to three units of the Courant number (0≤c≤3). Having good phase behavior, advection odd-order schemes are stable within a single computational cell (0≤c≤1). By contrast, even-order schemes are stable within two consecutive grid-cells (0≤c≤2), but exhibit poor dispersive properties. Stemming from the finding that considered higher-order upwind schemes (even, in particular) can be expressed as a linear combination of two lower-order ones (odd in this case), the best qualities of odd- and even-order algorithms can be blended within mixed-order accuracy schemes. To illustrate the idea, a Second-Order Reduced Dispersion (SORD) marching scheme and Fourth-Order Reduced Dispersion (FORD) upwind scheme are developed. Computational tests demonstrate a favorable performance of the schemes. In spite of the previous practice restricting usage of even-order upwind schemes (fourth-order in particular), they exhibit a potential to stand among popular algorithms of computational hydraulics.

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Hussain, Arafat, Zhoushun Zheng, and Eyaya Fekadie Anley. "Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method." Mathematics 8, no. 11 (2020): 1869. http://dx.doi.org/10.3390/math8111869.

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The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes.

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Dominguez, Hugo, Nicolas Riel, and Pierre Lanari. "Modelling chemical advection during magma ascent." Geoscientific Model Development 17, no. 16 (2024): 6105–22. http://dx.doi.org/10.5194/gmd-17-6105-2024.

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Abstract. Modelling magma transport requires robust numerical schemes for chemical advection. Current numerical schemes vary in their ability to be mass conservative, computationally efficient, and accurate. This study compares four of the most commonly used numerical schemes for advection: an upwind scheme, a weighted essentially non-oscillatory (WENO-5) scheme, a semi-Lagrangian (SL) scheme, and a marker-in-cell (MIC) method. The behaviour of these schemes is assessed using the passive advection of two different magmatic compositions. This is coupled in 2D with the temporal evolution of a melt anomaly that generates porosity waves. All algorithms, except the upwind scheme, are able to predict the melt composition with reasonable accuracy, but none of them is fully mass conservative. However, the WENO-5 scheme shows the best mass conservation. In terms of total running time and when multithreaded, the upwind, SL, and WENO-5 schemes show similar performance, while the MIC scheme is the slowest due to reseeding and removal of markers. The WENO-5 scheme has a reasonable total run time, has the best mass conservation, is easily parallelisable, and is therefore best suited for this problem.

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Scanlon, T. J., C. Carey, and S. M. Fraser. "SUCCA3D—An Alternative Scheme to Reduce False Diffusion in Three-Dimensional Flows." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 207, no. 5 (1993): 307–13. http://dx.doi.org/10.1243/pime_proc_1993_207_135_02.

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An alternative flow-oriented convection algorithm is presented which acts as a replacement for the first-order accurate UPWIND scheme in three-dimensional scalar transport. The scheme, formally titled SUCCA3D (skew upwind corner convection algorithm 3D), attempts to follow local streamlines, thus directly reducing the multi-dimensional false diffusion of the conserved scalar. In a standard benchmark test of pure convection across a three-dimensional cavity the SUCCA3D scheme was found to compare favourably with alternative schemes such as UPWIND and the higher-order QUICK scheme. The results highlight the potential of the SUCCA3D code for the reduction of three-dimensional false diffusion of a scalar variable in convection-dominated flows.

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van Trier, J., and W. W. Symes. "Upwind finite‐difference calculation of traveltimes." GEOPHYSICS 56, no. 6 (1991): 812–21. http://dx.doi.org/10.1190/1.1443099.

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Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite‐difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first‐arrival‐time field. The upwind finite‐difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first‐order upwind finite‐difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

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Castro Díaz, Manuel Jesús, Alexander Kurganov, and Tomás Morales de Luna. "Path-conservative central-upwind schemes for nonconservative hyperbolic systems." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 3 (2019): 959–85. http://dx.doi.org/10.1051/m2an/2018077.

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We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ``black-box’’ solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i) to show that the main drawback of the original central-upwind approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account and (ii) to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. We apply the new scheme to the Saint-Venant system with discontinuous bottom topography and two-layer shallow water system. Our numerical results illustrate the good performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution.

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Miura, Hiroaki. "An Upwind-Biased Conservative Advection Scheme for Spherical Hexagonal–Pentagonal Grids." Monthly Weather Review 135, no. 12 (2007): 4038–44. http://dx.doi.org/10.1175/2007mwr2101.1.

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Abstract A discrete form of the flux-divergence operator is developed to compute advection of tracers on spherical hexagonal–pentagonal grids. An upwind-biased advection scheme based on a piecewise linear approximation for one-dimensional regular grids is extended simply for spherical hexagonal–pentagonal grids. The distribution of a tracer over the upwind side of a cell face is linearly approximated using a nodal value and a gradient at a computational node on the upwind side. A piecewise linear approximation is relaxed to a local linear approximation, and the relaxation precludes the complicated conditional branching present in remapping schemes. Results from a cosine bell advection test show that the new scheme compares favorably with other upwind-biased schemes for spherical hexagonal–pentagonal grids.

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Zhang, Lin, Shu Yang Wang, and Guo Ling Niu. "Upwind Finite Element Method for Solving Radiative Heat Transfer in Graded Index Media." Advanced Materials Research 430-432 (January 2012): 1655–58. http://dx.doi.org/10.4028/www.scientific.net/amr.430-432.1655.

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Radiative transfer equation (RTE) in Cartesian coordinates can be considered as a special kind of convective-diffusive equation with strong convection characteristics. For this convection dominated problem, standard finite element solutions often suffer from spurious oscillations. To avoid this problem, the upwind finite element methods based on streamline upwind (SU) and streamline upwind Petrov-Galerkin (SUPG) schemes are developed to solve multidimensional radiative heat transfer in semitransparent uniform and graded index media. Comparison between these two upwind schemes on the solution of RTE is carried out. The SUPG scheme is demonstrated to be better than the SU scheme as far as solution accuracy is concerned and have good accuracy in solution of radiative heat transfer in semitransparent graded index media.

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Sukhinov, Alexander, Alexander Chistyakov, Elena Timofeeva, Alla Nikitina, and Yulia Belova. "The Construction and Research of the Modified “Upwind Leapfrog” Difference Scheme with Improved Dispersion Properties for the Korteweg–de Vries Equation." Mathematics 10, no. 16 (2022): 2922. http://dx.doi.org/10.3390/math10162922.

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This paper covers the construction and research of a scheme to solve the problem with nonlinear dispersion wave equations, described by the model Korteweg–de Vries equation. The article proposes approximating the equation based on improved “Upwind Leapfrog” schemes. Its difference operator is a linear combination of operators of the “Standard Leapfrog” and “Upwind Leapfrog” difference schemes, while the modified scheme is obtained from schemes with optimal weight coefficients. Combining certain values of the weight coefficients mutually compensates for approximation errors. In addition, the modified scheme acquires better properties compared with the original schemes. The results of test calculations of solutions of the nonlinear Korteweg–de Vries equation are presented, illustrating the advantages of the modified scheme.

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Baysal, Oktay. "Supercomputing of Supersonic Flows Using Upwind Relaxation and MacCormack Schemes." Journal of Fluids Engineering 110, no. 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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Sukhinov, Alexander, Alexander Chistyakov, Inna Kuznetsova, Elena Protsenko, and Yulia Belova. "Modified Upwind Leapfrog difference scheme." Computational Mathematics and Information Technologies 1, no. 1 (2020): 56–70. http://dx.doi.org/10.23947/2587-8999-2020-1-1-56-70.

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Zha, Ge-Cheng. "Low Diffusion Efficient Upwind Scheme." AIAA Journal 43, no. 5 (2005): 1137–40. http://dx.doi.org/10.2514/1.7726.

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Vuolo, Maria Raffaella, Laurent Menut, and Hélène Chepfer. "Impact of Transport Schemes on Modeled Dust Concentrations." Journal of Atmospheric and Oceanic Technology 26, no. 6 (2009): 1135–43. http://dx.doi.org/10.1175/2008jtecha1197.1.

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Abstract A sensitivity study is performed with the CHIMERE-DUST chemistry transport model in order to evaluate the modeled mineral dust spread due to the horizontal transport scheme accuracy. Three different schemes are implemented in the model: the simple first-order UPWIND scheme, the second-order Van Leer scheme, and the third-order parabolic piecewise method (PPM) scheme. The results showed that a large part of the uncertainty in dust modeling may be due to the transport scheme only. Compared to the PPM scheme, it is shown that, over a large domain encompassing western Africa and the North Atlantic, a significant increase in the dust plume extension is locally diagnosed (+25% with Van Leer and +48% with UPWIND) and linked to a decrease in the dust maxima (−17% with Van Leer and −32% with UPWIND) to PPM. Far from the sources, hourly surface concentration differences may be up to 30 μg m−3 in Europe, highlighting the high uncertainty of dust modeling for air quality use.

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Wu, Pan, Fei Chao, Dan Wu, Jianqiang Shan, and Junli Gou. "Implementation and Comparison of High-Resolution Spatial Discretization Schemes for Solving Two-Fluid Seven-Equation Two-Pressure Model." Science and Technology of Nuclear Installations 2017 (2017): 1–14. http://dx.doi.org/10.1155/2017/4252975.

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As compared to the two-fluid single-pressure model, the two-fluid seven-equation two-pressure model has been proved to be unconditionally well-posed in all situations, thus existing with a wide range of industrial applications. The classical 1st-order upwind scheme is widely used in existing nuclear system analysis codes such as RELAP5, CATHARE, and TRACE. However, the 1st-order upwind scheme possesses issues of serious numerical diffusion and high truncation error, thus giving rise to the challenge of accurately modeling many nuclear thermal-hydraulics problems such as long term transients. In this paper, a semi-implicit algorithm based on the finite volume method with staggered grids is developed to solve such advanced well-posed two-pressure model. To overcome the challenge from 1st-order upwind scheme, eight high-resolution total variation diminishing (TVD) schemes are implemented in such algorithm to improve spatial accuracy. Then the semi-implicit algorithm with high-resolution TVD schemes is validated on the water faucet test. The numerical results show that the high-resolution semi-implicit algorithm is robust in solving the two-pressure two-fluid two-phase flow model; Superbee scheme and Koren scheme give two highest levels of accuracy while Minmod scheme is the worst one among the eight TVD schemes.

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Starikovičius, V., and R. Čiegis. "ANALYSIS OF UPWIND AND HIGH‐RESOLUTION SCHEMES FOR SOLVING CONVECTION DOMINATED PROBLEMS IN POROUS MEDIA." Mathematical Modelling and Analysis 11, no. 4 (2006): 451–74. http://dx.doi.org/10.3846/13926292.2006.9637330.

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The conservation laws governing the multiphase flows in porous media are often convection‐dominated and have a steep fronts that require accurate resolution. Standard discretization methods of the convection terms do not perform well for such problems. The main aim of this work is to analyze the use of upwind and high‐ resolution schemes in such cases. First, we use a first differential approximation method to perform a theoretical analysis of a standard upwind approximation and different time stepping schemes for the linear hyperbolic equations in 1‐ and 2D. Next, we present a popular approach to reduce the amount of numerical diffusion introduced by upwind approximation ‐ high‐resolution schemes. We compare our implementation of one of the recently proposed central‐upwind schemes against the upwind schemes on several test problems based on Buckley‐Leverett equation and discuss the results. Finally, a parallel version of central‐upwind scheme in 2D is presented. It was implemented using our C++ library of parallel arrays ‐ ParSol.

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He, Shao Hua, Dong Yue Liu, and Da Li Tan. "Comparison of Various Spatial Discretization Schemes in Numerical Simulation for Ship Airwakes." Applied Mechanics and Materials 627 (September 2014): 63–68. http://dx.doi.org/10.4028/www.scientific.net/amm.627.63.

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For numerical simulation of ship airwake by CFD, based on the use of an unstructured grid, thek-εturbulence model and SIMPLE algorithm, the characteristic features of complex fluid flows eg recirculation zones and strong vortex fields in the aircarft operating region of a generic 3D frigate model was presented. The accuracy of the predication was checked by performing calculations on different grid sizes and comparing with wind-tunnel flow visualization data. A comparison of several typical spatial discretization schemes was performed.y+values were also tested. The general features of the flow predicted in this paper compare reasonably well with experimental data. However, CFD simulation produced a higher velocity in the vicinity of vortex zone when compared to experimental data. Obvious differences exist between results by first-order upwind scheme (power law scheme) and second-order upwind scheme (QUICK scheme, third-order MUSCL scheme). Second-order upwind scheme (QUICK scheme, third-order MUSCL scheme) are recommended for the CFD simulation of ship airwakes with a modest increase in computational cost.y+values from o (10) to o (1000) can all be accepted for the CFD simulation of ships (e.g., SFS1 ) with Reynolds number 108or more.

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Kim, Won, and Kun-Yeun Han. "Development of the Upwind McCormack Scheme." Journal of Korea Water Resources Association 38, no. 9 (2005): 727–36. http://dx.doi.org/10.3741/jkwra.2005.38.9.727.

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Falle, S. A. E. G., S. S. Komissarov, and P. Joarder. "A multidimensional upwind scheme for magnetohydrodynamics." Monthly Notices of the Royal Astronomical Society 297, no. 1 (1998): 265–77. http://dx.doi.org/10.1046/j.1365-8711.1998.01506.x.

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Roos, H. G. "A second order monotone upwind scheme." Computing 36, no. 1-2 (1986): 57–67. http://dx.doi.org/10.1007/bf02238192.

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Hwang, Yao-Hsin. "Upwind scheme for non-hyperbolic systems." Journal of Computational Physics 192, no. 2 (2003): 643–76. http://dx.doi.org/10.1016/j.jcp.2003.07.014.

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Sukhinov, Alexander, Alexander Chistyakov, Inna Kuznetsova, Yulia Belova, and Elena Rahimbaeva. "Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems." Mathematics 10, no. 19 (2022): 3564. http://dx.doi.org/10.3390/math10193564.

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Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

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Pereira, F. F., C. R. Fragoso Jr., C. B. Uvo, W. Collischonn, and D. M. L. Motta Marques. "Assessment of numerical schemes for solving the advection–diffusion equation on unstructured grids: case study of the Guaíba River, Brazil." Nonlinear Processes in Geophysics 20, no. 6 (2013): 1113–25. http://dx.doi.org/10.5194/npg-20-1113-2013.

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Abstract. In this work, a first-order upwind and a high-order flux-limiter schemes for solving the advection–diffusion equation on unstructured grids were evaluated. The numerical schemes were implemented as a module of an unstructured two-dimensional depth-averaged circulation model for shallow lakes (IPH-UnTRIM2D), and they were applied to the Guaíba River in Brazil. Their performances were evaluated by comparing mass conservation balance errors for two scenarios of a passive tracer released into the Guaíba River. The circulation model showed good agreement with observed data collected at four water level stations along the Guaíba River, where correlation coefficients achieved values up to 0.93. In addition, volume conservation errors were lower than 1% of the total volume of the Guaíba River. For all scenarios, the higher order flux-limiter scheme has been shown to be less diffusive than a first-order upwind scheme. Accumulated conservation mass balance errors calculated for the flux limiter reached 8%, whereas for a first-order upwind scheme, they were close to 18% over a 15-day period. Although both schemes have presented mass conservation errors, these errors are assumed negligible compared with kinetic processes such as erosion, sedimentation or decay rates.

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Yang, Qing. "The Upwind Finite Volume Element Method for Two-Dimensional Burgers Equation." Abstract and Applied Analysis 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/351619.

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A finite volume element method for approximating the solution to two-dimensional Burgers equation is presented. Upwind technique is applied to handle the nonlinear convection term. We present the semi-discrete scheme and fully discrete scheme, respectively. We show that the schemes are convergent to order one in space inL2-norm. Numerical experiment is presented finally to validate the theoretical analysis.

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Younes, Anis, Hussein Hoteit, Rainer Helmig, and Marwan Fahs. "A robust upwind mixed hybrid finite element method for transport in variably saturated porous media." Hydrology and Earth System Sciences 26, no. 20 (2022): 5227–39. http://dx.doi.org/10.5194/hess-26-5227-2022.

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Abstract. The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL). Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

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Moshiri, Mojtaba, and Mehrdad T. Manzari. "A comparative study of explicit high-resolution schemes for compositional simulations." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 1 (2019): 94–131. http://dx.doi.org/10.1108/hff-08-2017-0333.

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PurposeThis paper aims to numerically study the compositional flow of two- and three-phase fluids in one-dimensional porous media and to make a comparison between several upwind and central numerical schemes.Design/methodology/approachImplicit pressure explicit composition (IMPEC) procedure is used for discretization of governing equations. The pressure equation is solved implicitly, whereas the mass conservation equations are solved explicitly using different upwind (UPW) and central (CEN) numerical schemes. These include classical upwind (UPW-CLS), flux-based decomposition upwind (UPW-FLX), variable-based decomposition upwind (UPW-VAR), Roe’s upwind (UPW-ROE), local Lax–Friedrichs (CEN-LLF), dominant wave (CEN-DW), Harten–Lax–van Leer (HLL) and newly proposed modified dominant wave (CEN-MDW) schemes. To achieve higher resolution, high-order data generated by either monotone upstream-centered schemes for conservation laws (MUSCL) or weighted essentially non-oscillatory (WENO) reconstructions are used.FindingsIt was found that the new CEN-MDW scheme can accurately solve multiphase compositional flow equations. This scheme uses most of the information in flux function while it has a moderate computational cost as a consequence of using simple algebraic formula for the wave speed approximation. Moreover, numerically calculated wave structure is shown to be used as a tool for a priori estimation of problematic regions, i.e. degenerate, umbilic and elliptic points, which require applying correction procedures to produce physically acceptable (entropy) solutions.Research limitations/implicationsThis paper is concerned with one-dimensional study of compositional two- and three-phase flows in porous media. Temperature is assumed constant and the physical model accounts for miscibility and compressibility of fluids, whereas gravity and capillary effects are neglected.Practical implicationsThe proposed numerical scheme can be efficiently used for solving two- and three-phase compositional flows in porous media with a low computational cost which is especially useful when the number of chemical species increases.Originality/valueA new central scheme is proposed that leads to improved accuracy and computational efficiency. Moreover, to the best of authors knowledge, this is the first time that the wave structure of compositional model is investigated numerically to determine the problematic situations during numerical solution and adopt appropriate correction techniques.

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Sharma, Deepika, and Kavita Goyal. "Wavelet optimized upwind conservative method for traffic flow problems." International Journal of Modern Physics C 31, no. 06 (2020): 2050086. http://dx.doi.org/10.1142/s0129183120500862.

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Numerical schemes, namely, upwind nonconservative, upwind conservative, Lax–Friedrichs, Lax–Wendroff, MacCormack and Godunov are applied and compared on traffic flow problems. The best scheme, namely, upwind conservative is used for wavelet-optimized method using Daubechies wavelet for numerically solving the same traffic flow problems. Numerical results corresponding to the traffic flow problem with the help of wavelet-optimized, adaptive grid, upwind conservative method have been given. Moreover, the run time carried out by the developed technique have been compared to that of run time carried out by finite difference technique. It is observed that, in terms of run time, the proposed method performs better.

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Sharif, M. A. R., and A. A. Busnaina. "Evaluation and Comparison of Bounding Techniques for Convection-Diffusion Problems." Journal of Fluids Engineering 115, no. 1 (1993): 33–40. http://dx.doi.org/10.1115/1.2910109.

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The effects of bounding the skew upwind and the second-order upwind discretization schemes for the convection terms in convection-diffusion transport equations have been studied. Earlier studies indicated that these two schemes produce less numerical diffusion but introduce unacceptable numerical dispersion or oscillations in the solution if not bounded. A simplified analytical treatment exploring the reason for this behavior is presented. Two bounding techniques, the flux-corrected transport and the filtering remedy and methodology were evaluated. Test problems used in the evaluation are (i) one-dimensional convection of a rectangular pulse, (ii) transport of a scalar step in a uniform velocity field at an angle to the grid lines, (iii) Smith and Hutton problem, (iv) two-dimensional convection of a square scalar pulse in a uniform velocity field at an angle to the grid lines, and (v) two interacting parallel streams moving at an angle to the grid lines. The results indicate that the flux-corrected transport eliminates the oscillations in the solution without introducing any additional numerical diffusion when used with both schemes. The filtering remedy and methodology also eliminates the oscillation when used with the skew upwind scheme. This technique, however, is not effective in reducing the over-shoots when used with the second-order upwind scheme.

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Momoniat, E., M. M. Rashidi, and R. S. Herbst. "Numerical Investigation of Thin Film Spreading Driven by Surfactant Using Upwind Schemes." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/325132.

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Numerical solutions of a coupled system of nonlinear partial differential equations modelling the effects of surfactant on the spreading of a thin film on a horizontal substrate are investigated. A CFL condition is obtained from a von Neumann stability analysis of a linearised system of equations. Numerical solutions obtained from a Roe upwind scheme with a third-order TVD Runge-Kutta approximation to the time derivative are compared to solutions obtained with a Roe-Sweby scheme coupled to a minmod limiter and a TVD approximation to the time derivative. Results from both of these schemes are compared to a Roe upwind scheme and a BDF approximation to the time derivative. In all three cases high-order approximations to the spatial derivatives are employed on the interior points of the spatial domain. The Roe-BDF scheme is shown to be an efficient numerical scheme for capturing sharp changes in gradient in the free surface profile and surfactant concentration. Numerical simulations of an initial exponential free surface profile coupled with initial surfactant concentrations for both exogenous and endogenous surfactants are considered.

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Berthon, Christophe, Christian Klingenberg, and Markus Zenk. "An all Mach number relaxation upwind scheme." SMAI journal of computational mathematics 6 (April 24, 2020): 1–31. http://dx.doi.org/10.5802/smai-jcm.60.

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Parpia, Ijaz H., and Donna J. Michalek. "Grid-independent upwind scheme for multidimensional flow." AIAA Journal 31, no. 4 (1993): 646–51. http://dx.doi.org/10.2514/3.11598.

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Holstad, Astrid. "The Koren upwind scheme for variable gridsize." Applied Numerical Mathematics 37, no. 4 (2001): 459–87. http://dx.doi.org/10.1016/s0168-9274(00)00056-8.

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Bianchini, Stefano. "BV Solutions of the Semidiscrete Upwind Scheme." Archive for Rational Mechanics and Analysis 167, no. 1 (2003): 1–81. http://dx.doi.org/10.1007/s00205-002-0237-2.

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Barar, Farzad, and Seyed Esmail Razavi. "Pseudo-characteristic upwind scheme for incompressible flows." Advances in Mechanical Engineering 7, no. 12 (2015): 168781401561506. http://dx.doi.org/10.1177/1687814015615066.

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Chertock, Alina, Shaoshuai Chu, Michael Herty, Alexander Kurganov, and Mária Lukáčová-Medvid'ová. "Local characteristic decomposition based central-upwind scheme." Journal of Computational Physics 473 (January 2023): 111718. http://dx.doi.org/10.1016/j.jcp.2022.111718.

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Castro Diaz, Manuel Jesús, Yuanzhen Cheng, Alina Chertock, and Alexander Kurganov. "Solving Two-Mode Shallow Water Equations Using Finite Volume Methods." Communications in Computational Physics 16, no. 5 (2014): 1323–54. http://dx.doi.org/10.4208/cicp.180513.230514a.

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AbstractIn this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

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Kurganov, Alexander, Yongle Liu, and Vladimir Zeitlin. "Numerical dissipation switch for two-dimensional central-upwind schemes." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 3 (2021): 713–34. http://dx.doi.org/10.1051/m2an/2021009.

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We propose a numerical dissipation switch, which helps to control the amount of numerical dissipation present in central-upwind schemes. Our main goal is to reduce the numerical dissipation without risking oscillations. This goal is achieved with the help of a more accurate estimate of the local propagation speeds in the parts of the computational domain, which are near contact discontinuities and shears. To this end, we introduce a switch parameter, which depends on the distributions of energy in the x- and y-directions. The resulting new central-upwind is tested on a number of numerical examples, which demonstrate the superiority of the proposed method over the original central-upwind scheme.

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Zheng, Quan, Xue Zheng Li, and Yu Feng Liu. "Uniform Second-Order Hybrid Schemes on Bakhvalov-Shishkin Mesh for Quasi-Linear Convection-Diffusion Problems." Advanced Materials Research 871 (December 2013): 135–40. http://dx.doi.org/10.4028/www.scientific.net/amr.871.135.

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In this paper, we propose a class of hybrid difference schemes combining the central difference scheme and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh for solving quasi-linear singularly perturbed convection-diffusion boundary value problems. Point-wise second-order convergence uniform in the perturbation is proved clearly by using the-stability. The numerical experiments support the schemes and the uniform second-order estimate.

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Abedian, Rooholah. "High-Order Semi-Discrete Central-Upwind Schemes with Lax–Wendroff-Type Time Discretizations for Hamilton–Jacobi Equations." Computational Methods in Applied Mathematics 18, no. 4 (2018): 559–80. http://dx.doi.org/10.1515/cmam-2017-0031.

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AbstractA new fifth-order, semi-discrete central-upwind scheme with a Lax–Wendroff time discretization procedure for solving Hamilton–Jacobi (HJ) equations is presented. This is an alternative method for time discretization to the popular total variation diminishing (TVD) Runge–Kutta time discretizations. Unlike most of the commonly used high-order upwind schemes, the new scheme is formulated as a Godunov-type method. The new scheme is based on the flux Kurganov, Noelle and Petrova (KNP flux). The spatial discretization is based on a symmetrical weighted essentially non-oscillatory reconstruction of the derivative. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. Various numerical experiments are performed to demonstrate the accuracy and stability properties of the new method. As a result, comparing with other fifth-order schemes for HJ equations, the major advantage of the new scheme is more cost effective for certain problems while the new method exhibits smaller errors without any increase in the complexity of the computations.

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Lin, San-Yin, Sheng-Chang Shih, and Jen-Jiun Hu. "Dissipation Improvement of MUSCL Scheme for Computational Aeroacoustics." Journal of Mechanics 17, no. 1 (2001): 39–47. http://dx.doi.org/10.1017/s1727719100002409.

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ABSTRACTAn upwind finite-volume scheme is studied for solving the solutions of two dimensional Euler equations. It based on the MUSCL (Monotone Upstream Scheme for Conservation Laws) approach with the Roe approximate Riemann solver for the numerical flux evaluation. First, dissipation and dispersion relation, and group velocity of the scheme are derived to analyze the capability of the proposed scheme for capturing physical waves, such as acoustic, entropy, and vorticity waves. Then the scheme is greatly enhanced through a strategy on the numerical dissipation to effectively handle aeroacoustic computations. The numerical results indicate that the numerical dissipation strategy allows that the scheme simulates the continuous waves, such as sound and sine waves, at fourth-order accuracy and captures the discontinuous waves, such a shock wave, sharply as well as most of upwind schemes do. The tested problems include linear wave convection, propagation of a sine-wave packet, propagation of discontinuous and sine waves, shock and sine wave interaction, propagation of acoustic, vorticity, and density pulses in an uniform freestream, and two-dimensional traveling vortex in a low-speed freestream.

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Zijlema, Marcel. "Physics-Capturing Discretizations for Spectral Wind-Wave Models." Fluids 6, no. 2 (2021): 52. http://dx.doi.org/10.3390/fluids6020052.

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This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes for the transport of wave action in geographical space. The discussion addresses the derivation of these schemes from a different perspective. A mathematical framework for mimetic discretizations based on discrete calculus is utilized herein. A key feature of this algebraic approach is that the process of exact discretization is segregated from the process of interpolation, the latter typically involved in constitutive relations. This can help gain insight into the performance characteristics of the discretization method. On this basis, we conclude that the upwind finite difference scheme captures the wave action flux conservation exactly, which is a plus for wave shoaling. In addition, we provide a justification for the intrinsic low accuracy of the vertex-centred upwind finite volume scheme, due to the physically inaccurate but common flux constitutive relation, and we propose an improvement to overcome this drawback. Finally, by way of a comparative demonstration, a few test cases is introduced to establish the ability of the considered methods to capture the relevant physics on unstructured triangular meshes.

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Fei, Tan Jeff, and Puay How Tion. "The Performance of CIP Scheme in Solving Advection Equation." International Journal on Engineering Technology and Infrastructure Development 1, no. 1 (2024): 104–10. http://dx.doi.org/10.3126/injet-indev.v1i1.67941.

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Constrained interpolation profile (CIP) approach is reviewed in this paper. This research provides a performance comparison between CIP and upwind scheme, a different general method for solving advection equations. The performance of numerical method will be judged based on its accuracy in solving advection equation. The simulation results indicate that CIP scheme advection is more accurate than Upwind scheme advection due to its minor numerical deviation from exact solution.

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Wu, Conghai, Sujuan Yang, and Ning Zhao. "A Fifth-Order Low-Dissipative Conservative Upwind Compact Scheme Using Centered Stencil." Advances in Applied Mathematics and Mechanics 6, no. 06 (2014): 830–48. http://dx.doi.org/10.4208/aamm.2013.m-s3.

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AbstractIn this paper, a conservative fifth-order upwind compact scheme using centered stencil is introduced. This scheme uses asymmetric coefficients to achieve the upwind property since the stencil is symmetric. Theoretical analysis shows that the proposed scheme is low-dissipative and has a relatively large stability range. To maintain the convergence rate of the whole spatial discretization, a proper non-periodic boundary scheme is also proposed. A detailed analysis shows that the spatial discretization implemented with the boundary scheme proposed by Pirozzoli [J. Comput. Phys., 178 (2001), pp. 81–117] is approximately fourth-order. Furthermore, a hybrid methodology, coupling the compact scheme with WENO scheme, is adopted for problems with discontinuities. Numerical results demonstrate the effectiveness of the proposed scheme.

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Madaliev, Murodil, Jahongir Orzimatov, Zokhidjon Abdulkhaev, Olimjon Esonov, and Mirzohid Mirzaraximov. "Several different ways to increase the accuracy of the numerical solution of the first order wave equation." BIO Web of Conferences 84 (2024): 02032. http://dx.doi.org/10.1051/bioconf/20248402032.

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The article presents several different ways to increase the accuracy of the numerical solution of differential equations. The comparison of schemes with different accuracy for the first order wave equation problem is presented. The schemes of the first order of upwind scheme, the second order of accuracy of McCormack, the third order of accuracy of Warming–Cutler–Lomax, and the scheme of the fourth order of accuracy of Abarbanel–Gotlieb–Turkel were applied. The condensed computational grids for the McCormack scheme are used, and the results using an adaptive grid for the McCormack scheme were compared.

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Zhang, Guiyong, Da Hui, Da Li, Li Zou, Shengchao Jiang, and Zhi Zong. "A New TVD Scheme for Gradient Smoothing Method Using Unstructured Grids." International Journal of Computational Methods 17, no. 03 (2019): 1850132. http://dx.doi.org/10.1142/s0219876218501323.

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An improved [Formula: see text]-factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity.

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Serguini, Abdelhafid, Sanae Jelti, and Abdelmajid El Hajaji. "Well-Balanced conservative central upwind scheme for solving the dam-break flow problem over erodible bed." Boletim da Sociedade Paranaense de Matemática 42 (May 22, 2024): 1–16. http://dx.doi.org/10.5269/bspm.66942.

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This work deals with the numerical solution of dam-break flow over an erodible bed. The mathematical model is a combination of the shallow water, the transport diffusion and the bed morphology change equations. The system is solved by a well-Balanced central upwind scheme with conservative property. Several tests are illustrated in order to validate the accuracy and the performance of the model. A comparison of central upwind scheme and Roe scheme is presented.

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Ginting, Bobby, and Ralf-Peter Mundani. "Comparison of Shallow Water Solvers: Applications for Dam-Break and Tsunami Cases with Reordering Strategy for Efficient Vectorization on Modern Hardware." Water 11, no. 4 (2019): 639. http://dx.doi.org/10.3390/w11040639.

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We investigate in this paper the behaviors of the Riemann solvers (Roe and Harten-Lax-van Leer-Contact (HLLC) schemes) and the Riemann-solver-free method (central-upwind scheme) regarding their accuracy and efficiency for solving the 2D shallow water equations. Our model was devised to be spatially second-order accurate with the Monotonic Upwind Scheme for Conservation Laws (MUSCL) reconstruction for a cell-centered finite volume scheme—and be temporally fourth-order accurate using the Runge–Kutta fourth-order method. Four benchmark cases of dam-break and tsunami events dealing with highly-discontinuous flows and wet–dry problems were simulated. To this end, we applied a reordering strategy for the data structures in our code supporting efficient vectorization and memory access alignment for boosting the performance. Two main features are pointed out here. Firstly, the reordering strategy employed has enabled highly-efficient vectorization for the three solvers investigated on three modern hardware (AVX, AVX2, and AVX-512), where speed-ups of 4.5–6.5× were obtained on the AVX/AVX2 machines for eight data per vector while on the AVX-512 machine we achieved a speed-up of up to 16.7× for 16 data per vector, all with singe-core computation; with parallel simulations, speed-ups of up to 75.7–121.8× and 928.9× were obtained on AVX/AVX2 and AVX-512 machines, respectively. Secondly, we observed that the central-upwind scheme was able to outperform the HLLC and Roe schemes 1.4× and 1.25×, respectively, by exhibiting similar accuracies. This study would be useful for modelers who are interested in developing shallow water codes.

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