Relevant bibliographies by topics / Modified Crank­Nicolson Method

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Academic literature on the topic 'Modified Crank­Nicolson Method'

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Published: 7 June 2025
Last updated: 16 July 2025

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Journal articles on the topic "Modified Crank­Nicolson Method"

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Tejaskumar, C. Sharma, P. Pathak Shreekant, and Trivedi Gargi. "Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations." Indian Journal of Science and Technology 17, no. 10 (2024): 924–31. https://doi.org/10.17485/IJST/v17i10.1776.

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Abstract <strong>Objectives:</strong>&nbsp;This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method.&nbsp;<strong>Methods:</strong>&nbsp;Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and &gamma; = 0.1. Conducted extensive testing on various partial differential equations.<strong>&nbsp;Findings:</strong>&nbsp;Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superiority in accuracy.<strong>&nbsp;Novelty:</strong>&nbsp;The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. <strong>Keywords</strong>: Crank&shy;Nicolson Method, Modified Crank&shy;Nicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software
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Sharma, Tejaskumar C., Shreekant P. Pathak, and Gargi Trivedi. "Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations." Indian Journal Of Science And Technology 17, no. 10 (2024): 924–31. http://dx.doi.org/10.17485/ijst/v17i10.1776.

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Objectives: This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. Methods: Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations. Findings: Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superiority in accuracy. Novelty: The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. Keywords: Crank­Nicolson Method, Modified Crank­Nicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software
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WANI, SACHIN S. "Modified Crank Nicolson Type Method for Burgers Equation." Journal of Ultra Scientist of Physical Sciences Section A 28, no. 5 (2016): 259–77. http://dx.doi.org/10.22147/jusps-a/280505.

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Shi, Yao, Xiaozhen Liu, and Zhenyu Wang. "An Efficient Structure-Preserving Scheme for the Fractional Damped Nonlinear Schrödinger System." Fractal and Fractional 9, no. 5 (2025): 328. https://doi.org/10.3390/fractalfract9050328.

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This paper introduces a highly accurate and efficient conservative scheme for solving the nonlocal damped Schrödinger system with Riesz fractional derivatives. The proposed approach combines the Fourier spectral method with the Crank–Nicolson time-stepping scheme. To begin, the original equation is reformulated into an equivalent system by introducing a new variable that modifies both energy and mass. The Fourier spectral method is employed to achieve high spatial accuracy in this semi-discrete formulation. For time discretization, the Crank–Nicolson scheme is applied, ensuring conservation of the modified energy and mass in the fully discrete system. Numerical experiments validate the scheme’s precision and its ability to preserve conservation properties.
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Kurniawan, A. R., E. Suaebah, and Z. A. I. Supardi. "Modified Crank-Nicolson method for solving the time-dependent Schrödinger equation." Journal of Physics: Conference Series 2900, no. 1 (2024): 012009. https://doi.org/10.1088/1742-6596/2900/1/012009.

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Abstract The time-dependent Schrödinger equation is fundamental to quantum mechanics, describing the evolution of quantum systems. This study introduces a novel numerical approach that extends and modifies the Crank-Nicholson method to solve the time-dependent Schrödinger equation. This method is designed to enhance both the precision and performance of computational results, especially in the context of quantum tunneling phenomena and applications that require high accuracy over long periods. To assess the precision and performance of the modified Crank-Nicolson method, we benchmark it against two well-known numerical methods: the Runge-Kutta method and the Split-Step Fourier method. Quantum tunneling is selected as the case study due to its relevance in quantum physics applications. Comparative analysis is based on numerical error metrics. The proposed method can be applied to other physical phenomena, expanding its potential application in physics research and technology industries.
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Arora, Geeta, Shubham Mishra, Homan Emaifar, and Masoumeh Khademi. "Numerical Simulation and Dynamics of Burgers’ Equation Using the Modified Cubic B-Spline Differential Quadrature Method." Discrete Dynamics in Nature and Society 2023 (March 29, 2023): 1–8. http://dx.doi.org/10.1155/2023/5102374.

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In the present work, a numerical approach using the Crank–Nicolson scheme along with the modified cubic B-spline differential quadrature (CN-MCDQ) method is proposed to find the numerical approximations to Burgers’ equation. After applying the well-known Crank–Nicolson technique, Burgers’ equation is solved in this study by using the differential quadrature approach to approximate the derivatives that lead to a system of equations to be solved. When compared to other methods for obtaining numerical solutions, the proposed method is shown to be efficient and easy to implement while still providing accurate results. The obtained results are in agreement with the earlier available approaches and are even better in comparison in terms of less domain partition. Three test problems were used to evaluate the methodology, and the results are tabulated and graphically shown below.
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Wu, Chunya, Xinlong Feng, and Lingzhi Qian. "A Second-Order Crank–Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction." Entropy 24, no. 11 (2022): 1512. http://dx.doi.org/10.3390/e24111512.

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In this paper, we construct a fully discrete and decoupled Crank–Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank–Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method.
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Alanazi, A. A., Sultan Z. Alamri, S. Shafie, and Shazirawati Mohd Puzi. "Crank-Nicolson Scheme for Solving the Modified Nonlinear Schrodinger Equation." International Journal of Numerical Methods for Heat & Fluid Flow 31, no. 8 (2021): 2789–817. http://dx.doi.org/10.1108/hff-10-2020-0677.

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Purpose The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme. Design/methodology/approach The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities. Findings The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients. Originality/value The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.
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Castillo, P. E., and S. A. Gómez. "Conservación de invariantes de la ecuación de Schrödinger no lineal por el método LDG." Revista Mexicana de Física E 64, no. 1 (2018): 52. http://dx.doi.org/10.31349/revmexfise.64.52.

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Conservation of the energy and the Hamiltonian of a general non linear Schr¨odinger equation is analyzed for the finite element method “Local Discontinuous Galerkin” spatial discretization. Conservation of the discrete analogue of these quantities is also proved for the fully discrete problem using the modified Crank-Nicolson method as time marching scheme. The theoretical results are validated on a series of problemsfor different nonlinear potentials.
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Ashyralyev, Allaberen, and Ali Sirma. "Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation." Discrete Dynamics in Nature and Society 2009 (2009): 1–15. http://dx.doi.org/10.1155/2009/584718.

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The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy -modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.
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Dissertations / Theses on the topic "Modified Crank­Nicolson Method"

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Ferreira, Sabrina dos Santos 1984. "Estudo do método dos elementos finitos de mínimos quadrados - LSFEM para resolução da equação de convecção - difusão bidimensional." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/265744.

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Orientador: Luiz Felipe Mendes de Moura<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica<br>Made available in DSpace on 2018-08-28T11:40:36Z (GMT). No. of bitstreams: 1 Ferreira_SabrinadosSantos_M.pdf: 4964592 bytes, checksum: 48a2af83f98b244951f9db97d31a68cc (MD5) Previous issue date: 2015<br>Resumo: O objetivo deste trabalho foi o estudo da distribuição de temperatura em um domínio retangular, para tal foi resolvida a Equação de Convecção - Difusão Bidimensional via Método dos Elementos Finitos de Mínimos Quadrados - Least Squares Finite Element Method (LSFEM). Para discretização espacial foi utilizado elementos bidimensionais quadráticos, nesse caso os elementos quadriláteros com oito nós foram escolhidos. A discretização temporal nos casos transientes foi aproximada via Método de Crank - Nicolson. Para a o obtenção da matriz do elemento e o vetor do lado direito a quadratura de Gauss - Legendre foi empregada. A solução do sistema algébrico resultante foi obtida a partir do Método dos Gradientes Conjugados, um dos métodos iterativos mais eficientes na resolução de sistemas lineares quando a matriz é simétrica, esparsa e definida positiva, sendo essas características resultantes da formulação via LSFEM. É apresentada a formulação matemática do problema, a metodologia empregada na solução. Para a obtenção dos resultados um código em linguagem C foi implementado, por fim são apresentados os resultados obtidos, as conclusões e sugestões para trabalhos futuros<br>Abstract: The objective of this work was the study of temperature distribution in a rectangular domain, for that was solved Equation Convection - Diffusion Two-Dimensional via Least Squares Finite Element Method - LSFEM. For spatial discretization was used two-dimensional quadratic elements, in this case the quadrilateral elements with eight nodes were chosen. The time discretization in transient cases was approached via Crank - Nicolson Method. To obtain the element matrix and vector right side of the Gauss - Legendre quadrature was employed. The solution of resulting algebraic system was obtained using Conjugate Gradient Method, one of the most efficient iterative methods for solving linear sistenas when the matrix is ??symmetric, and positive definite sparse, and these resulting characteristics of the formulation via LSFEM. The mathematical formulation of the problem, the methodology used in the solution is shown. To obtain the results a code in C language was implemented, finally is presented results, conclusions and suggestions for future work<br>Mestrado<br>Termica e Fluidos<br>Mestra em Engenharia Mecânica
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Carmona, Tabares Paulo Cesar 1976. "Impacto do sedimento sobre espécies que interagem = modelagem e simulações de bentos na Enseada Potter." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307271.

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Orientador: João Frederico da Costa Azevedo Meyer<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica<br>Made available in DSpace on 2018-08-21T04:55:31Z (GMT). No. of bitstreams: 1 CarmonaTabares_PauloCesar_D.pdf: 24565019 bytes, checksum: 8ebe9aed1d258a0712f49e9711f8d107 (MD5) Previous issue date: 2012<br>Resumo: Neste trabalho, construímos um modelo matemático para avaliar as conjecturas existentes acerca do impacto que tem o material inorgânico particulado (sedimento) nas populações bentônicas predominantes na Enseada Potter. Na construção do modelo são utilizadas informações do fenômeno, proporcionadas pelas pesquisas permanentes na região de estudo. Como resultado, logramos comprovar mediante simulações numéricas, o efeito que produz o sedimento na distribuição e abundância das espécies do substrato marinho, constatando neste ecossistema particular as consequências do aquecimento global nessa parte da região antártica. A modelagem é feita com um sistema de equações diferenciais parciais não- lineares sobre um domínio bidimensional irregular (descritiva da região original), o qual é discretizado nas variáveis espaciais por elementos finitos de primeira ordem e na variável temporal pelo Método de Crank-Nicolson. A resolução do sistema não-linear resultante é aproximada através de um método preditor-corretor cuja solução aproximada é visualizada e valorada qualitativamente usando gráficos evolutivos obtidos por simulações em ambiente MATLAB<br>Abstract: In this work, we built a mathematical model to evaluate existing conjectures about the impact that inorganic particulate material (sediment) has upon predominating benthic populations in Potter Cove. For the mathematical model, phenomena information was that provided by permanent researches in the study area. As a result, by means of numerical simulations, we were able to confirm the effect of sediment over distribution and abundance for species of marine substrate, verifying in this particular ecosystem, the effects of global warming in this specific Antarctic region. Modeling is done with a system of nonlinear partial differential equations over an irregular two-dimensional domain (descriptive of the original region), which is discretized in the spatial variables by first order finite elements and in the time variable by Crank-Nicolson. The resolution of the resulting nonlinear system is approximated by a predictor-corrector method and the solution is displayed and qualitatively valorized using evolutive graphics, obtain in a MATLAB environment<br>Doutorado<br>Matematica Aplicada<br>Doutor em Matemática Aplicada
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Wang, Mianzhi. "Numerical Analysis of Transient Teflon Ablation with a Domain Decomposition Finite Volume Implicit Method on Unstructured Grids." Digital WPI, 2012. https://digitalcommons.wpi.edu/etd-theses/284.

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This work investigates numerically the process of Teflon ablation using a finite-volume discretization, implicit time integration and a domain decomposition method in three-dimensions. The interest in Teflon stems from its use in Pulsed Plasma Thrusters and in thermal protection systems for reentry vehicles. The ablation of Teflon is a complex process that involves phase transition, a receding external boundary where the heat flux is applied, an interface between a crystalline and amorphous (gel) phase and a depolymerization reaction which happens on and beneath the ablating surface. The mathematical model used in this work is based on a two-phase model that accounts for the amorphous and crystalline phases as well as the depolymerization of Teflon in the form of an Arrhenius reaction equation. The model accounts also for temperature-dependent material properties, for unsteady heat inputs and boundary conditions in 3D. The model is implemented in 3D domains of arbitrary geometry with a finite volume discretization on unstructured grids. The numerical solution of the transient reaction-diffusion equation coupled with the Arrhenius-based ablation model advances in time using implicit Crank-Nicolson scheme. For each time step the implicit time advancing is decomposed into multiple sub-problems by a domain decomposition method. Each of the sub-problems is solved in parallel by Newton-Krylov non-linear solver. After each implicit time-advancing step, the rate of ablation and the fraction of depolymerized material are updated explicitly with the Arrhenius-based ablation model. After the computation, the surface of ablation front and the melting surface are recovered from the scalar field of fraction of depolymerized material and the fraction of melted material by post-processing. The code is verified against analytical solutions for the heat diffusion problem and the Stefan problem. The code is validated against experimental data of Teflon ablation. The verification and validation demonstrates the ability of the numerical method in simulating three dimensional ablation of Teflon.
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Lindqvist, Petter. "Validering av solida temperaturer i FDS genom jämförelse mot FE-beräkningar." Thesis, Luleå tekniska universitet, Institutionen för samhällsbyggnad och naturresurser, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-79973.

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FDS (Fire Dynamics Simulator) använder en version av Navier-Stokes ekvationerna för att göra noggranna beräkningar av värme- och gastransport genom brandbelastade utrymmen. Utvecklarna av programmet arbetar kontinuerligt med att validera det allteftersom nya funktioner tillförs för att öka noggrannheten och bredda tillämpningsområdena. Väldigt lite av detta arbete fokuserar dock på FDS:s konduktionsmodell, den endimensionella Crank-Nicolson metoden. Det här examensarbetet ämnar därför undersöka noggrannheten i FDS:s konduktionsmodell genom jämförelse mot beräkningar med FEM (Finita elementmetoden). En FDS-modell skapades för att tillåta undersökning av en vägg och dess randvillkor med så liten påverkan från andra faktorer som möjligt. Detta för att skapa en kontrollerad omgivning som enkelt kunde replikeras i efterföljande FE-beräkningar av det konduktiva värmeflödet genom den solida obstruktionen. Tre väggar (10 cm betong, 20 cm betong och 1 mm stål) vardera med tre randvillkor (Exposed, Void och Insulated) utsattes för tre temperaturer (100 °C, 500 °C och 1000 °C) vilket ger 27 FDS simuleringar. Den adiabatiska yttemperaturen mättes i varje simulering och användes som indata till motsvarande FE-beräkningar. Resultatet påvisade inga signifikanta motsägelser vad gäller randvillkoren, med tillräcklig tid för termisk penetrering påverkade de den resulterande temperaturen som väntat. Undantaget var en mindre avvikelse i stålväggarna som utsattes för 100 °C och 500 °C med randvillkoren Exposed och Void där FDS aningen underskattade temperaturen relativt FE-beräkningarna. Gastemperaturerna i gridcellerna närmast väggen visade sig vara opålitliga. De tenderade att genomgå substantiella fluktuationer, troligen som ett resultat av hur FDS hanterar diskretiseringen av icke-solida volymer för Navier-Stokes beräkningarna. Dessa fluktuationer påverkade dock inte de resulterande solida temperaturerna eftersom medelgastemperaturen var korrekt. FDS påvisades även ha en tendens att aningen överskatta yttemperaturen under de första minuterna av simuleringarna relativt FE-beräkningarna. Temperaturerna från de två beräkningsmetoderna konvergerade dock efter några få minuter i samtliga tester. Dessa avvikelser ansågs ha för liten påverkan på de solida temperaturerna för att påvisa onoggrannhet i FDS. Därmed drogs slutsatsen att FDS:s beräkningar av temperaturer i solida material är tillräckligt noggranna inom dessa avgränsningar.<br>FDS (Fire Dynamics Simulator) uses a version of the Navier-Stokes equations to make accurate calculations of heat and gas flow through enclosures exposed to fire. The developers of FDS have, and continue to, validate it as new features get added in an attempt to increase its accuracy and broaden its potential applications. However, little of this effort is focused on FDS’ conductive heat transfer model, based on the one-dimensional Crank-Nicolson method. Thus, this study aims to test the accuracy of FDS’ conduction model by comparing it to calculations using FEM (Finite Element Method). FDS simulations were created so as to facilitate the study of a wall and its boundary conditions with as little interference from other factors as possible. This to create a controlled environment which easily could be replicated in the subsequent FE-calculations of the conductive heat flow through the solid obstructions. Three different walls (10 cm concrete, 20 cm concrete and 1 mm steel), each with the three different boundary conditions for the rear surface (Exposed, Void and Insulated), were exposed to three different temperatures (100 °C, 500 °C and 1000 °C) for a total of 27 FDS simulations. The adiabatic surface temperature was measured in each simulation in FDS and used as input for the corresponding FE-calculations. The results showed no clear inconsistencies in the boundary conditions, given enough time for thermal penetration they affected the resulting temperatures as expected. Save a slight deviation in the steel walls exposed to 100 °C and 500 °C with boundary conditions Exposed and Void where FDS slightly underestimated the temperature relative to the FE-calculations. The gas temperatures in the grid cells closest to the wall were found to be unreliable as they tended to undergo substantial fluctuations, likely as a result of how FDS handles the discretization of non-solid space for the Navier-Stokes calculations. These fluctuations were however not found to affect the solid temperatures as the mean gas temperature was accurate. FDS was also found to have a tendency to slightly overestimate the surface temperature in the first few minutes of the simulations relative to the FE-calculations. Though the resulting temperatures from the two methods converged after a few minutes at most in all tests. These deviations were considered to have too minor an impact on the solid temperature to justify claims of inaccuracy in FDS. Thus, the general conclusion of this study is that FDS’ predictions of solid phase temperatures are sufficiently accurate within these delimitations.
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Agrawal, Devanshu. "A Numerical Model for Nonadiabatic Transitions in Molecules." Digital Commons @ East Tennessee State University, 2014. https://dc.etsu.edu/honors/193.

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In molecules, electronic state transitions can occur via quantum coupling of the states. If the coupling is due to the kinetic energy of the molecular nuclei, then electronic transitions are best represented in the adiabatic frame. If the coupling is instead facilitated through the potential energy of the nuclei, then electronic transitions are better represented in the diabatic frame. In our study, we modeled these latter transitions, called ``nonadiabatic transitions.'' For one nuclear degree of freedom, we modeled the de-excitation of a diatomic molecule. For two nuclear degrees of freedom, we modeled the de-excitation of an ethane-like molecule undergoing cis-trans isomerization. For both cases, we studied the dependence of the de-excitation on the nuclear configuration and potential energy of the molecule. We constructed a numerical model to solve the time-dependent Schr\"{o}dinger Equation for two coupled wave functions. Our algorithm takes full advantage of the sparseness of the numerical system, leading to a final set of equations that is solved recursively using nothing more than the Tridiagonal Algorithm. We observed that the most effective de-excitation occurred when the molecule transitioned from a stable equilibrium configuration to an unstable equilibrium configuration. This same mechanism is known to drive fast electronic transitions in the adiabatic frame. We concluded that while the adiabatic and diabatic frames are strongly opposed physically, the mathematical mechanism driving electronic transitions in the two frames is in some sense the same.
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Rouf, Hasan. "Unconditionally stable finite difference time domain methods for frequency dependent media." Thesis, University of Manchester, 2010. https://www.research.manchester.ac.uk/portal/en/theses/unconditionally-stable-finite-difference-time-domain-methods-for-frequency-dependent-media(50e4adf1-d1e4-4ad2-ab2d-70188fb8b7b6).html.

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The efficiency of the conventional, explicit finite difference time domain (FDTD)method is constrained by the upper limit on the temporal discretization, imposed by the Courant–Friedrich–Lewy (CFL) stability condition. Therefore, there is a growing interest in overcoming this limitation by employing unconditionally stable FDTD methods for which time-step and space-step can be independently chosen. Unconditionally stable Crank Nicolson method has not been widely used in time domain electromagnetics despite its high accuracy and low anisotropy. There has been no work on the Crank Nicolson FDTD (CN–FDTD) method for frequency dependent medium. In this thesis a new three-dimensional frequency dependent CN–FDTD (FD–CN–FDTD) method is proposed. Frequency dependency of single–pole Debye materials is incorporated into the CN–FDTD method by means of an auxiliary differential formulation. In order to provide a convenient and straightforward algorithm, Mur’s first-order absorbing boundary conditions are used in the FD–CN–FDTD method. Numerical tests validate and confirm that the FD–CN–FDTD method is unconditionally stable beyond the CFL limit. The proposed method yields a sparse system of linear equations which can be solved by direct or iterative methods, but numerical experiments demonstrate that for large problems of practical importance iterative solvers are to be used. The FD–CN–FDTD sparse matrix is diagonally dominant when the time-stepis near the CFL limit but the diagonal dominance of the matrix deteriorates with the increase of the time-step, making the solution time longer. Selection of the matrix solver to handle the FD–CN–FDTD sparse system is crucial to fully harness the advantages of using larger time-step, because the computational costs associated with the solver must be kept as low as possible. Two best–known iterative solvers, Bi-Conjugate Gradient Stabilised (BiCGStab) and Generalised Minimal Residual (GMRES), are extensively studied in terms of the number of iteration requirements for convergence, CPU time and memory requirements. BiCGStab outperforms GMRES in every aspect. Many of these findings do not match with the existing literature on frequency–independent CN–FDTD method and the possible reasons for this are pointed out. The proposed method is coded in Fortran and major implementation techniques of the serial code as well as its parallel implementation in Open Multi-Processing (OpenMP) are presented. As an application, a simulation model of the human body is developed in the FD–CN–FDTD method and numerical simulation of the electromagnetic wave propagation inside the human head is shown. Finally, this thesis presents a new method modifying the frequency dependent alternating direction implicit FDTD (FD–ADI–FDTD) method. Although the ADI–FDTD method provides a computationally affordable approximation of the CN–FDTD method, it exhibits a loss of accuracy with respect to the CN-FDTD method which may become severe for some practical applications. The modified FD–ADI–FDTD method can improve the accuracy of the normal FD–ADI–FDTD method without significantly increasing the computational costs.
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Joubert, Dominique. "Numerical methods for pricing American put options under stochastic volatility / Dominique Joubert." Thesis, North-West University, 2013. http://hdl.handle.net/10394/10202.

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The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation.<br>MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013
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Akman, Tugba. "Discontinuous Galerkin Methods For Time-dependent Convection Dominated Optimal Control Problems." Master's thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613394/index.pdf.

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Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensional problem. For one-dimensional unconstrained problem, Newton-Conjugate Gradient method with Armijo line-search. For two-dimensional control constrained problem, active-set method is applied. A priori error estimates are derived for full discretized optimal control problem. Numerical results for one and two-dimensional distributed optimal control problems for diffusion convection equations with boundary layers confirm the predicted orders derived by a priori error estimates.
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Vergez, Guillaume. "Méthodes numériques avec des éléments finis adaptatifs pour la simulation de condensats de Bose-Einstein." Thesis, Normandie, 2017. http://www.theses.fr/2017NORMR014/document.

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Le phénomène de condensation d’un gaz de bosons lorsqu’il est refroidi à zéro degrés Kelvin futdécrit par Einstein en 1925 en s’appuyant sur des travaux de Bose. Depuis lors, de nombreux physiciens,mathématiciens et numériciens se sont intéressés au condensat de Bose-Einstein et à son caractère superfluide. Nous proposons dans cette étude des méthodes numériques ainsi qu’un code informatique pour la simulation d’un condensat de Bose-Einstein en rotation. Le principal modèle mathématique décrivant ce phénomène physique est une équation de Schrödinger présentant une non-linéarité cubique,découverte en 1961 : l’équation de Gross-Pitaevskii (GP). En nous appuyant sur le logiciel FreeFem++,nous nous servons d’une discrétisation spatiale en éléments-finis pour résoudre numériquement cette équation. Une méthode d’adaptation du maillage à la solution et l’utilisation d’éléments-finis d’ordre deux nous permet de résoudre finement le problème et d’explorer des configurations complexes en deux ou trois dimensions d’espace. Pour sa version stationnaire, nous avons développé une méthode de gradient de Sobolev ou une méthode de point intérieur implémentée dans la librairie Ipopt. Pour sa version instationnaire, nous utilisons une méthode de Time-Splitting combinée à un schéma de Crank-Nicolson ou une méthode de relaxation. Afin d’étudier la stabilité dynamique et thermodynamique d’un état stationnaire, le modèle de Bogoliubov-de Gennes propose une linéarisation de l’équation de Gross-Pitaevskii autour de cet état. Nous avons élaboré une méthode permettant de résoudre ce système aux valeurs et vecteurs propres, basée sur un algorithme de Newton ainsi que sur la méthode d’Arnoldi implémentée dans la librairie Arpack<br>The phenomenon of condensation of a boson gas when cooled to zero degrees Kelvin was described by Einstein in 1925 based on work by Bose. Since then, many physicists, mathematicians and digitizers have been interested in the Bose-Einstein condensate and its superfluidity. We propose in this study numerical methods as well as a computer code for the simulation of a rotating Bose-Einstein condensate.The main mathematical model describing this phenomenon is a Schrödinger equation with a cubic nonlinearity, discovered in 1961: the Gross-Pitaevskii (GP) equation. By using the software FreeFem++ and a finite elements spatial discretization we solve this equation numerically. The mesh adaptation to the solution and the use of finite elements of order two allow us to solve the problem finely and to explore complex configurations in two or three dimensions of space. For its stationary version, we have developed a Sobolev gradient method or an internal point method implemented in the Ipopt library. .For its unsteady version, we use a Time-Splitting method combined with a Crank-Nicolson scheme ora relaxation method. In order to study the dynamic and thermodynamic stability of a stationary state,the Bogoliubov-de Gennes model proposes a linearization of the Gross-Pitaevskii equation around this state. We have developed a method to solve this eigenvalues and eigenvector system, based on a Newton algorithm as well as the Arnoldi method implemented in the Arpack library
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Váňa, Martin. "Časově závislé řešení dvourozměrných rozptylových problémů v kvantové mechanice." Master's thesis, 2012. http://www.nusl.cz/ntk/nusl-305089.

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The scope of this thesis is in the time-dependent formulation of the two dimensional model of resonant electron-diatomic molecule collisions in the range of low energies. In its time independent form the model was previously numerically solved without the Born-Oppenheimer approximation with use of modern tools such as the finite element method with discrete variable representation (FEM-DVR) or exterior complex scaling (ECS). Within the scope of this model we numerically solve the evolution problem, with use of the Crank-Nicolson method and the Padé approximation. Later we evaluate the cross section of the elastic and some inelastic processes with the correlation function approach. At last we make a comparison of the evolution and the cross sections to time dependent formulation of the local complex potential approximation of the electron-molecule collisions.
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Books on the topic "Modified Crank­Nicolson Method"

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B, Fulton Robert, and Langley Research Center, eds. Concurrent implementation of the Crank-Nicolson method for heat transfer analysis. National Aeronautics and Space Administration, Langley Research Center, 1985.

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Boudreau, Joseph F., and Eric S. Swanson. Continuum dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198708636.003.0019.

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The theory and application of a variety of methods to solve partial differential equations are introduced in this chapter. These methods rely on representing continuous quantities with discrete approximations. The resulting finite difference equations are solved using algorithms that stress different traits, such as stability or accuracy. The Crank-Nicolson method is described and extended to multidimensional partial differential equations via the technique of operator splitting. An application to the time-dependent Schrödinger equation, via scattering from a barrier, follows. Methods for solving boundary value problems are explored next. One of these is the ubiquitous fast Fourier transform which permits the accurate solution of problems with simple boundary conditions. Lastly, the finite element method that is central to modern engineering is developed. Methods for generating finite element meshes and estimating errors are also discussed.
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Book chapters on the topic "Modified Crank­Nicolson Method"

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Kwok, Felix. "A Parallel Crank–Nicolson Predictor-Corrector Method for Many Subdomains." In Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05789-7_77.

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Marciniak, Andrzej. "An Interval Version of the Crank-Nicolson Method – The First Approach." In Applied Parallel and Scientific Computing. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-28145-7_12.

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Faragó, István. "Qualitative Analysis of the Crank-Nicolson Method for the Heat Conduction Equation." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-00464-3_5.

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Zhou, Jun, and Min Xiong. "An Extrapolation Method of Crank-Nicolson Finite Difference Scheme for Distributed Control Equation." In Lecture Notes in Electrical Engineering. Springer London, 2013. http://dx.doi.org/10.1007/978-1-4471-4850-0_23.

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Victor, Franklin, John J. H. Miller, and Valarmathi Sigamani. "Convergence of the Crank-Nicolson Method for a Singularly Perturbed Parabolic Reaction-Diffusion System." In Springer Proceedings in Mathematics & Statistics. Springer India, 2016. http://dx.doi.org/10.1007/978-81-322-3598-9_5.

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Kumar, Sachin. "Crank-Nicolson Quasi-Wavelet Method for the Numerical Solution of Variable-Order Time-Space Riesz Fractional Reaction-Diffusion Equation." In Applications of Fractional Calculus to Modeling in Dynamics and Chaos. Chapman and Hall/CRC, 2022. http://dx.doi.org/10.1201/9781003006244-16.

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Jankowska, Malgorzata A. "An Interval Finite Difference Method of Crank-Nicolson Type for Solving the One-Dimensional Heat Conduction Equation with Mixed Boundary Conditions." In Applied Parallel and Scientific Computing. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-28145-7_16.

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Li, Ru, Na Li, Fei Chen, and Xin Zhao. "Dynamics Study of Mechanism with Clearance Based on Adams." In Lecture Notes in Mechanical Engineering. Springer Nature Singapore, 2025. https://doi.org/10.1007/978-981-97-7887-4_56.

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Abstract In view of the influence of the motion pair gap on the dynamic characteristics of the mechanism, considering the gap size and the driving load, an improved nonlinear contact collision model is proposed, the friction with the modified Kulun force friction model. Movement pair gap with mass vector rod equivalent, using discrete analysis method for collision between the motion, the crank slider mechanism as an example, the clearance mechanism model in Adams, using the algorithm of unilateral impact function to calculate the collision force, quantitative analysis of the dynamic characteristics of the mechanism, verify the consistency of the simulation results and theoretical value, provides a theoretical basis for the design and analysis of the clearance mechanism.
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Benkhaldoun, Fayssal, and Abdallah Bradji. "A New Analysis for a Super-Convergence Result in the Divergence Norm for Lowest Order Raviart–Thomas Mixed Finite Elements Combined with the Crank–Nicolson Method Applied to One Dimensional Parabolic Equations." In Springer Proceedings in Mathematics & Statistics. Springer Nature Switzerland, 2023. http://dx.doi.org/10.1007/978-3-031-40864-9_11.

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Rizea, M. "Improved Crank-Nicolson method applied to quantum tunnelling." In Recent Progress in Computational Sciences and Engineering. CRC Press, 2019. http://dx.doi.org/10.1201/9780429070655-111.

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Conference papers on the topic "Modified Crank­Nicolson Method"

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Kawahara, Akira, and Jun Shibayama. "Subgridding FDTD Method Based on the Iterated Crank-Nicolson Scheme." In 2024 Asia-Pacific Microwave Conference (APMC). IEEE, 2024. https://doi.org/10.1109/apmc60911.2024.10867527.

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Abdullatif, Firas, and Dean Wang. "Implementation and Assessment of the Crank-Nicolson Method for the Monte Carlo Multilevel Kinetics Module." In International Conference on Physics of Reactors (PHYSOR 2024). American Nuclear Society, 2024. http://dx.doi.org/10.13182/physor24-43529.

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Seraj, Sabet, Amin Fereidooni, and Anant Grewal. "A Comparison of Strong and Weak Coupling Schemes for Computational Aeroelasticity in OpenFOAM." In ASME 2018 5th Joint US-European Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/fedsm2018-83292.

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Two coupling schemes for fluid-structure interaction using the OpenFOAM structural solver sixDoF Rigid Body Motion are developed. The first scheme is developed by modifying the baseline leapfrog weak coupling scheme to minimize the lag between the fluid and structural solvers. The second is a strong coupling scheme based on the Crank-Nicolson method. The two newly implemented schemes and the baseline are compared through the aeroelastic simulation of a NACA 64A010 airfoil and the Benchmark Supercritical Wing. The aeroelastic solutions obtained using the newly implemented schemes exhibit significantly lower sensitivity to changes in time step size compared to the baseline weak coupling scheme. The modified weak coupling and strong coupling schemes perform comparably for the cases studied.
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Baker, Mark, and Budimir Rosic. "1D ANALYTIC AND NUMERICAL ANALYSIS OF MULTILAYER LAMINATES AND THIN FILM HEAT TRANSFER GAUGES." In GPPS Xi'an21. GPPS, 2022. http://dx.doi.org/10.33737/gpps21-tc-346.

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The impulse response method is widely used for heat transfer analysis in turbomachinery applications. Traditionally, the 1D method assumes a linear time invariant, isotropic, semi-infinite block and does not accurately model the behaviour of laminated materials. This paper evaluates the error introduced by the single layer assumption and outlines the required modifications for multilayer analysis. The analytic solution for an N layer, semi-infinite laminate is presented. Adapted multilayer basis functions are derived for the impulse response method and used to evaluate the impact of uniform, isotropic assumptions. A numerical solution to the laminate problem is also presented. A penta-diagonal inversion algorithm, for a modified Crank-Nicolson scheme, is evaluated for fast stable implementation of multilayer simulation. The scheme shows comparable performance to the impulse response, whilst removing the requirement for linear time invariance. The methods are demonstrated in the case of analysing a thin film gauge, used in laboratory analysis of heat transfer in a turbine nozzle guide vane. Thin film gauge manufacturing techniques have advanced significantly in recent years. Advanced multilayer constructions are now used however, post-processing commonly relies on outdated single layer methods. This paper provides a universal methodology, required to analyse modern-day multilayer heat transfer measurements.
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Baker, Mark, and Budimir Rosic. "1D analytic and numerical analysis of thin film heat transfer gauges and infra-red cameras in non-planar applications." In GPPS Hong Kong24. GPPS, 2023. http://dx.doi.org/10.33737/gpps23-tc-194.

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The impulse response method is widely used for heat transfer analysis in turbomachinery applications. Traditionally, the 1D method assumes a: linear time invariant, isotropic, semi-infinite block with planar surfaces and does not accurately model the true geometric behaviour. This paper evaluates the error introduced by the planar assumption and outlines the required modifications for accurate freeform surface analysis. Adapted cylindrical basis functions are defined for the impulse response method and used to evaluate the impact of the 1D planar assumption. The analytic solutions for both convex and concave surfaces are presented. A penta-diagonal algorithm, for a modified numerical Crank-Nicolson scheme, is also evaluated for fast stable implementation of curved geometry simulation. The scheme shows comparable performance to the impulse response, whilst removing the requirement for linear time invariance. The new methods are demonstrated in the case of aerothermal analysis for heat transfer in a turbine nozzle guide vane. A 3D ANSYS simulation is used as a benchmark and further highlights the importance of the curvature effects. The methods are extended using curvature mapping for infrared camera data, enabling direct 3D thermal simulation or the fast calculation of freeform curvature. This paper defines the methodology to analyse heat transfer measurements on non-planar geometry.
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Yee, Chan Choon, and Azwani Alias. "Solving Burgers equation numerically using Crank-Nicolson method." In PROCEEDINGS OF SCIEMATHIC 2020. AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0053460.

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Dagang Wu and Ji Chen. "Perfectly matched layer for Crank-Nicolson (CN) FDTD method." In IEEE Antennas and Propagation Society Symposium, 2004. IEEE, 2004. http://dx.doi.org/10.1109/aps.2004.1329737.

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Kawahara, Akira, Jun Shibayama, Junji Yamauchi, and Hisamatsu Nakano. "Numerical Dispersion Analysis of the Iterated Crank-Nicolson-Based FDTD Method." In XXXVth URSI General Assembly and Scientific Symposium. URSI – International Union of Radio Science, 2023. http://dx.doi.org/10.46620/ursigass.2023.0653.swea6857.

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Guiffaut, C., A. Reineix, and B. Pecqueux. "Local Crank-Nicolson procedure for short thin wire in the FDTD method." In 2012 IEEE Antennas and Propagation Society International Symposium and USNC/URSI National Radio Science Meeting. IEEE, 2012. http://dx.doi.org/10.1109/aps.2012.6348867.

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Blasik, Marek. "A Generalized Crank-Nicolson Method for the Solution of the Subdiffusion Equation." In 2018 23rd International Conference on Methods & Models in Automation & Robotics (MMAR). IEEE, 2018. http://dx.doi.org/10.1109/mmar.2018.8485908.

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