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A Crank-Nicolson Difference Scheme for Generalized Rosenau Equation 被引量:1
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作者 胡劲松 王玉兰 徐友才 《Journal of Southwest Jiaotong University(English Edition)》 2010年第3期254-259,共6页
Much effort has been devoted to researching the common Rosenau equation, but the numerical method of it has not been studied. In this paper, a conservative Crank-Nicolson difference scheme for an initial-boundary valu... Much effort has been devoted to researching the common Rosenau equation, but the numerical method of it has not been studied. In this paper, a conservative Crank-Nicolson difference scheme for an initial-boundary value problem of the generalized Rosenau equation is proposed. Existence and uniqueness of numerical solutions are derived. By method of discrete energy, the second order convergence and stability are discussed. Numerical examples demonstrate the theoretical results. 展开更多
关键词 Generalized Rosenau equation difference scheme Conservative rule CONVERGENCE Stability
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Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts
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作者 Dinshaw S.Balsara Deepak Bhoriya +1 位作者 Chi-Wang Shu Harish Kumar 《Communications on Applied Mathematics and Computation》 EI 2024年第2期907-962,共56页
Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of ... Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages. 展开更多
关键词 Hyperbolic PDEs Numerical schemes Non-conservative products Stiff source terms Finite difference WENO
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Analysis of Extended Fisher-Kolmogorov Equation in 2D Utilizing the Generalized Finite Difference Method with Supplementary Nodes
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作者 Bingrui Ju Wenxiang Sun +1 位作者 Wenzhen Qu Yan Gu 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第10期267-280,共14页
In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolso... In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem. 展开更多
关键词 Generalized finite difference method nonlinear extended Fisher-Kolmogorov equation crank-nicolson scheme
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New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy 被引量:1
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作者 Jun Zhu Jianxian Qiu 《Communications on Applied Mathematics and Computation》 2023年第1期64-96,共33页
In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbol... In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation laws.For the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more wider.The more we use wider candidate stencils,the bigger the probability of discontinuities lies in all stencils.Therefore,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils.By the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(J.Comput.Phys.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth regions.These new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL number.Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes. 展开更多
关键词 Finite difference Mapped WENO scheme Mapping function Mapped nonlinear weight Unequal-sized stencil Extreme example
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Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations
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作者 Dechao Gao Zeshan Qiu +1 位作者 Lizan Wang Jianxin Li 《Journal of Applied Mathematics and Physics》 2024年第4期1089-1100,共12页
The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference oper... The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective. 展开更多
关键词 crank-nicolson Quasi-Compact scheme Fractional Advection-Diffusion Equations NONLINEAR Stability and Convergence
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AWeighted Average Finite Difference Scheme for the Numerical Solution of Stochastic Parabolic Partial Differential Equations
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作者 Dumitru Baleanu Mehran Namjoo +1 位作者 Ali Mohebbian Amin Jajarmi 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第5期1147-1163,共17页
In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic fi... In the present paper,the numerical solution of It?type stochastic parabolic equation with a timewhite noise process is imparted based on a stochastic finite difference scheme.At the beginning,an implicit stochastic finite difference scheme is presented for this equation.Some mathematical analyses of the scheme are then discussed.Lastly,to ascertain the efficacy and accuracy of the suggested technique,the numerical results are discussed and compared with the exact solution. 展开更多
关键词 Itoequation stochastic process finite difference scheme stability and convergence CONSISTENCY
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Finite Difference Approach for Estimating the Thermal Conductivity by 6-point Crank-Nicolson Scheme
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作者 苏亚欣 杨翔翔 《Journal of Donghua University(English Edition)》 EI CAS 2005年第2期45-49,共5页
Based on inverse heat conduction theory, a theoretical model using 6-point Crank-Nicolson finite difference scheme was used to calculate the thermal conductivity from temperature distribution, which can be measured ex... Based on inverse heat conduction theory, a theoretical model using 6-point Crank-Nicolson finite difference scheme was used to calculate the thermal conductivity from temperature distribution, which can be measured experimentally. The method is a direct approach of second-order and the key advantage of the present method is that it is not required a priori knowledge of the functional form of the unknown thermal conductivity in the calculation and the thermal parameters are estimated only according to the known temperature distribution. Two cases were numerically calculated and the influence of experimental deviation on the precision of this method was discussed. The comparison of numerical and analytical results showed good agreement. 展开更多
关键词 inverse heat conduction thermal conductivity 6-point finite difference scheme
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A-high-order Accuraqcy Implicit Difference Scheme for Solving the Equation of Parabolic Type 被引量:7
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作者 马明书 王肖凤 《Chinese Quarterly Journal of Mathematics》 CSCD 2000年第2期94-97,共4页
In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(... In this paper,a implicit difference scheme is proposed for solving the equation of one_dimension parabolic type by undetermined paameters.The stability condition is r=αΔt/Δx 2 1/2 and the truncation error is o(Δt 4+Δx 4) It can be easily solved by double sweeping method. 展开更多
关键词 equation of one_dimension parabolic type high_order accuracy implicit difference scheme
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CONSERVATIVE DIFFERENCE SCHEME BASED ON NUMERICAL ANALYSIS FOR NONLINEAR SCHRDINGER EQUATION WITH WAVE OPERATOR 被引量:2
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作者 王廷春 张鲁明 陈芳启 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI 2006年第2期87-93,共7页
A new conservative finite difference scheme is presented based on the numerical analysis for an initialboundary value problem of a class of Schroedinger equation with the wave operator. The scheme can be linear and im... A new conservative finite difference scheme is presented based on the numerical analysis for an initialboundary value problem of a class of Schroedinger equation with the wave operator. The scheme can be linear and implicit or explicit based on the parameter choice. The initial value after discretization has second-order accuracy that is consistent with the scheme accuracy. The existence and the uniqueness of the difference solution are proved. Based on the priori estimates and an inequality about norms, the stability and the convergence of difference solutions with the second-order are proved in the energy norm. Experimental results demonstrate the efficiency of the new scheme. 展开更多
关键词 Schroedinger equation difference scheme CONSERVATION existence and uniqueness of solution CONVERGENCE
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An Explicit Difference Scheme with High Accuracy and Branching Stability for Solving Parabolic Partial Differential Equation 被引量:4
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作者 马明书 王肖凤 《Chinese Quarterly Journal of Mathematics》 CSCD 2000年第4期98-103,共6页
This paper presents an explicit difference scheme with accuracy and branching stability for solving onedimensional parabolic type equation by the method of undetermined parameters and its truncation error is O(△t4+△... This paper presents an explicit difference scheme with accuracy and branching stability for solving onedimensional parabolic type equation by the method of undetermined parameters and its truncation error is O(△t4+△x4). The stability condition is r=a△t/△x2<1/2. 展开更多
关键词 parabolic type equation explicit difference scheme high accuracy branching stability truncation er
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A Family of High_order Accuracy Explicit Difference Schemes for Solving 2-D Parabolic Partial Differential Equation 被引量:4
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作者 任宗修 陈贞忠 王肖凤 《Chinese Quarterly Journal of Mathematics》 CSCD 2002年第3期57-61,共5页
A family of high_order accuracy explicit difference schemes for solving 2_dimension parabolic P.D.E. are constructed. Th e stability condition is r=Δt/Δx 2=Δt/Δy 2【1/2 and the truncation err or is O(Δt 3+Δx... A family of high_order accuracy explicit difference schemes for solving 2_dimension parabolic P.D.E. are constructed. Th e stability condition is r=Δt/Δx 2=Δt/Δy 2【1/2 and the truncation err or is O(Δt 3+Δx 4). 展开更多
关键词 D parabolic P.D.E high_order accuracy explic it difference scheme
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HIGH RESOLUTION POSITIVITY-PRESERVING DIFFERENCE SCHEMES FOR TWO DIMENSIONAL EULER EQUATIONS
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作者 赵宁 张虎 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI 2000年第2期163-168,共6页
A class of high resolution positivity preserving Boltzmann type difference schemes for one and two dimensional Euler equations is studied. First, the relation between Boltzmann and Euler equations is analyzed. By usi... A class of high resolution positivity preserving Boltzmann type difference schemes for one and two dimensional Euler equations is studied. First, the relation between Boltzmann and Euler equations is analyzed. By using a kind of special interpolation, the high resolution Boltzmann type difference scheme is constructed. Finally, numerical tests show that the schemes are effective and useful. 展开更多
关键词 Euler equation Boltzmann equation finite difference scheme positivity preserving
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High-Order Bound-Preserving Finite Difference Methods for Multispecies and Multireaction Detonations 被引量:1
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作者 Jie Du Yang Yang 《Communications on Applied Mathematics and Computation》 2023年第1期31-63,共33页
In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical ... In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical reaction,say zi,is between 0 and 1,withΣz_(i)=1.Due to the lack of maximum-principle,most of the previous bound-preserving technique cannot be applied directly.To preserve those bounds,we will use the positivity-preserving technique to all the zi'is and enforceΣz_(i)=1 by constructing conservative schemes,thanks to conservative time integrations and consistent numerical fluxes in the system.Moreover,detonation is an extreme singular mode of flame propagation in premixed gas,and the model contains a significant stiff source.It is well known that for hyperbolic equations with stiff source,the transition points in the numerical approximations near the shocks may trigger spurious shock speed,leading to wrong shock position.Intuitively,the high-order weighted essentially non-oscillatory(WENO)scheme,which can suppress oscillations near the discontinuities,would be a good choice for spatial discretization.However,with the nonlinear weights,the numerical fluxes are no longer“consistent”,leading to nonconservative numerical schemes and the bound-preserving technique does not work.Numerical experiments demonstrate that,without further numerical techniques such as subcell resolutions,the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme. 展开更多
关键词 Weighted essentially non-oscillatory scheme Finite difference method Stiff source DETONATIONS Bound-preserving CONSERVATIVE
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Stability of Difference Schemes with Intrinsic Parallelism for Quasilinear Parabolic Systems 被引量:10
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作者 Zhou Yulin and Yuan GuangweiLaboratory of Computational PhysicsCentre for Nonlinear Studies(Institute of Applied Physics and Computational MathematicsP. O. Box 8009, Boiling, 100088, China) 《Wuhan University Journal of Natural Sciences》 CAS 1996年第Z1期579-592,共14页
In this paper, the first boundary problem of quasilinear parabolic system of second order is studied by the finite difference method with intrinsic parallelism. for the problem, the stability of the difference schemes... In this paper, the first boundary problem of quasilinear parabolic system of second order is studied by the finite difference method with intrinsic parallelism. for the problem, the stability of the difference schemes with intrinsic parallelism are justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete data of the original problem, without assuming the existence of the smooth solutions for the origillal problem. 展开更多
关键词 difference scheme Intrinsic parallelism Quasilinear 'parabolic systems Stability.
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A CONSERVATIVE DIFFERENCE SCHEME FOR CONSERVATIVE DIFFERENTIAL EQUATION WITH PERIODIC BOUNDARY 被引量:3
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作者 CAI Xin(蔡新) 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2001年第10期1210-1215,共6页
The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the ... The conservative form and singular perturbed ordinary differential equation with periodic boundary value problem were studied, and a conservative difference scheme was constructed. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, it is proved that the scheme converges uniformly to the solution of differential equation with order one. 展开更多
关键词 conservative equation singular perturbation periodic boundary conservative difference scheme uniform convergence
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The Stability Research for the Finite Difference Scheme of a Nonlinear Reaction-diffusion Equation 被引量:6
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作者 XU Chen-mei 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第2期222-227,共6页
In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite differ... In the article, the fully discrete finite difference scheme for a type of nonlinear reaction-diffusion equation is established. Then the new function space is introduced and the stability problem for the finite difference scheme is discussed by means of variational approximation method in this function space. The approach used is of a simple characteristic in gaining the stability condition of the scheme. 展开更多
关键词 reaction-diffusion equation finite difference scheme stability research variational approximation method
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A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES 被引量:1
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作者 涂国华 袁湘江 +1 位作者 夏治强 呼振 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2006年第6期765-772,共8页
A new method was proposed for constructing total variation diminishing (TVD) upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can e... A new method was proposed for constructing total variation diminishing (TVD) upwind schemes in conservation forms. Two limiters were used to prevent nonphysical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD (CTVD) schemes were tested, one is thirdorder accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a twodimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities. 展开更多
关键词 high-order difference schemes compact schemes TVD schemes shock- vortex shock-boundary
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Second-order difference scheme for a nonlinear model of wood drying process
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作者 姜明杰 孙志忠 《Journal of Southeast University(English Edition)》 EI CAS 2006年第4期582-588,共7页
A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordin... A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result. 展开更多
关键词 wood drying process model nonlinear differential equation difference scheme method of reduction of order STABILITY CONVERGENCE
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A High-order Accuracy Explicit Difference Scheme with Branching Stability for Solving Higher-dimensional Heat-conduction Equation 被引量:3
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作者 MA Ming-shu MA Ju-yi +1 位作者 GU Shu-min ZHU Lin-lin 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第3期446-452,共7页
A high-order accuracy explicit difference scheme for solving 4-dimensional heatconduction equation is constructed. The stability condition is r = △t/△x^2 = △t/△y^2 = △t/△z^2 = △t/△w^2 〈 3/8, and the truncatio... A high-order accuracy explicit difference scheme for solving 4-dimensional heatconduction equation is constructed. The stability condition is r = △t/△x^2 = △t/△y^2 = △t/△z^2 = △t/△w^2 〈 3/8, and the truncation error is O(△t^2 + △x^4). 展开更多
关键词 heat-conduction equation explicit difference scheme high-order accuracy branching stability
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A SIMPLE PROOF OF STABILITY AND CONVERGENCE IN L_2 FOR SOME DIFFERENCE SCHEMES AND THEIR EXTRAPOLATION METHOD FOR PARABOLIC EQUATIONS 被引量:1
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作者 孙志忠 《Journal of Southeast University(English Edition)》 EI CAS 1994年第2期1-6,共6页
ASIMPLEPROOFOFSTABILITYANDCONVERGENCEINL_2FORSOMEDIFFERENCESCHEMESANDTHEIREXTRAPOLATIONMETHODFORPARABOLICEQU... ASIMPLEPROOFOFSTABILITYANDCONVERGENCEINL_2FORSOMEDIFFERENCESCHEMESANDTHEIREXTRAPOLATIONMETHODFORPARABOLICEQUATIONSSunZhizhong... 展开更多
关键词 simpld PROOF Wabelic equatiuns difference schemeS STABILITY convergenee EXTRAPOLATION METHOD
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