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Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes 认领 引用
1
作者 M. S. Ismail Fakhirah Alotaibi 《Applied Mathematics》 2020年第11期1139-1161,共23页
In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two ... In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well. 展开更多
关键词 MKdV Equation Pade Approximation Nonlinear Numerical Schemes Linearly Implicit Schemes Fixed Point Method Interaction of Solitons
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Semi-Implicit Scheme to Solve Allen-Cahn Equation with Different Boundary Conditions 认领 引用
2
作者 Banan Alqanawi Musa Adam Aigo 《American Journal of Computational Mathematics》 2023年第1期122-135,共14页
The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part o... The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently. 展开更多
关键词 Semi-Implicit Schemes Allen-Cahn Equations Finite Difference Sparse System Jacobi Fixed Point Gauss-Seidel
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Fully implicit improved discrete velocity method for diatomic gas in all flow regimes 认领 引用 被引量：2
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作者 Linchang Han Liming Yang +3 位作者 Zhihui Li Yanan Ma Ruoyu Li Hao Dong 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2026年第2期45-63,共19页
The improved discrete velocity method(IDVM),a multiscale kinetic approach,solves both the Boltzmann model equation and the macroscopic governing equations.It maintains the simplicity of the conventional discrete veloc... The improved discrete velocity method(IDVM),a multiscale kinetic approach,solves both the Boltzmann model equation and the macroscopic governing equations.It maintains the simplicity of the conventional discrete velocity method while significantly enhancing accuracy and efficiency in the continuum flow regime.However,previous implementations of IDVM were limited to monatomic gases,rendering it inapplicable under real atmospheric conditions dominated by diatomic gases.In this paper,the fully implicit IDVM is extended to the Boltzmann-Rykov model equation to simulate multiscale gas flows involving the rotational non-equilibrium effects of diatomic molecules.This approach incorporates macroscopic governing equations to predict the equilibrium distribution function,enabling the fully implicit discretization of the BoltzmannRykov model equation and ensuring rapid convergence.Numerical simulations are performed for several cases,including the one-dimensional nitrogen shock tube,two-dimensional flows around a flat plate and a blunt circular cylinder,and threedimensional supersonic flow over a sphere.The results of the present algorithm show remarkable alignment with those from similar kinetic algorithms and experimental data,demonstrating its ability to simulate non-equilibrium diatomic gas flows.Relative to the conventional semi-implicit discrete velocity method and implicit kinetic methods with a macroscopic forecasting technique,the proposed approach attains varying degrees of acceleration. 展开更多
关键词 Rotational non-equilibrium effect Improved discrete velocity method Rykov model Fully implicit scheme
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A MONOTONE COMPACT IMPLICIT SCHEME FOR NONLINEAR REACTION-DIFFUSION EQUATIONS 认领 引用 被引量：5
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作者 Yuanming Wang Benyu Guo 《Journal of Computational Mathematics》 SCIE EI 2008年第2期123-148,共26页
A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations.An accelerated monotone iterative method... A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations.An accelerated monotone iterative method for the resulting discrete problem is presented.The sequence of iteration converges monotonically to the unique solution of the discrete problem,and the convergence rate is either quadratic or nearly quadratic,depending on the property of the nonlinear reaction.The numerical results illustrate the high accuracy of the proposed scheme and the rapid convergence rate of.the iteration. 展开更多
关键词 Nonlinear reaction-diffusion equation Monotone compact implicit scheme High accuracy Monotone iteration Rapid convergence rate.
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A unified implicit scheme for kinetic model equations. Part I. Memory reduction technique 认领 引用 被引量：9
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作者 Songze Chen Chuang Zhang +1 位作者 Lianhua Zhu Zhaoli Guo 《Science Bulletin》 SCIE EI CAS CSCD 2017年第2期119-129,共11页
A memory reduction technique is proposed for solving stationary kinetic model equations. As implied by an integral solution of the stationary kinetic equation, a velocity distribution function can be reconstructed fro... A memory reduction technique is proposed for solving stationary kinetic model equations. As implied by an integral solution of the stationary kinetic equation, a velocity distribution function can be reconstructed from given macroscopic variables. Based on this fact, we propose a technique to reconstruct distribution function at discrete level, and employ it to develop an implicit numerical method for kinetic equations. The new implicit method only stores the macroscopic quantities which appear in the collision term, and does not store the distribution functions. As a result, enormous memory requirement for solving kinetic equations is totally relieved. Several boundary conditions, such as, inlet, outlet and isothermal boundaries, are discussed. Some numerical tests demonstrate the validity and efficiency of the technique.The new implicit solver provides nearly identical solution as the explicit kinetic solver, while the memory requirement is on the same order as the Navier–Stokes solver. 展开更多
关键词 Implicit scheme Kinetic equation Memory reduction
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Analysis of an Implicit Finite Difference Scheme for Time Fractional Diffusion Equation 认领 引用 被引量：1
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作者 MA Yan 《Chinese Quarterly Journal of Mathematics》 2016年第1期69-81,共13页
Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order tim... Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper. 展开更多
关键词 time fractional diffusion equation finite difference approximation implicit scheme stability convergence effectiveness
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A-high-order Accuraqcy Implicit Difference Scheme for Solving the Equation of Parabolic Type 认领 引用 被引量：9
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作者 马明书 王肖凤 《Chinese Quarterly Journal of Mathematics》 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(Δt... 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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Unconditional Stability and Fourth-Order Convergence of a Two-Step Time Split Explicit/Implicit Scheme for Two-Dimensional Nonlinear Unsteady Convection-Diffusion-Reaction Equation 认领 引用
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作者 Eric Ngondiep Ali H.Tedjani 《Advances in Applied Mathematics and Mechanics》 SCIE 2024年第6期1381-1409,共29页
This paper deals with an efficient two-step time split explicit/implicit scheme applied to a two-dimensional nonlinear unsteady convection-diffusionreaction equation.The computational cost of the new algorithm at each... This paper deals with an efficient two-step time split explicit/implicit scheme applied to a two-dimensional nonlinear unsteady convection-diffusionreaction equation.The computational cost of the new algorithm at each time level is equivalent to solving a pentadiagonalmatrix equation with strictly dominant diagonal elements.Such a bandwidth matrix can be easily inverted using the Gaussian Decomposition and the corresponding linear system should be solved by the back substitutionmethod.The proposed approach is unconditionally stable,temporal second-order accuracy and fourth-order convergence in space.These results suggest that the developed technique is faster and more efficient than a large class of numerical methods studied in the literature for the considered initial-boundary value problem.Numerical experiments are carried out to confirm the theoretical analysis and to demonstrate the performance of the constructed numerical scheme. 展开更多
关键词 2D nonlinear unsteady convection-diffusion-reaction equation explicit method implicit scheme two-step time split fourth-order explicit/implicit approach unconditional stability error estimates
High-order implicit discontinuous Galerkin schemes for unsteady compressible Navier–Stokes equations 认领 引用 被引量：4
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作者 Jiang Zhenhua Yan Chao Yu Jian 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2014年第6期1384-1389,共6页
Efficient solution techniques for high-order temporal and spatial discontinuous Galerkin（DG） discretizations of the unsteady Navier–Stokes equations are developed. A fourth-order implicit Runge–Kutta（IRK） scheme... Efficient solution techniques for high-order temporal and spatial discontinuous Galerkin（DG） discretizations of the unsteady Navier–Stokes equations are developed. A fourth-order implicit Runge–Kutta（IRK） scheme is applied for the time integration and a multigrid preconditioned GMRES solver is extended to solve the nonlinear system arising from each IRK stage. Several modifications to the implicit solver have been considered to achieve the efficiency enhancement and meantime to reduce the memory requirement. A variety of time-accurate viscous flow simulations are performed to assess the resulting high-order implicit DG methods. The designed order of accuracy for temporal discretization scheme is validate and the present implicit solver shows the superior performance by allowing quite large time step to be used in solving time-implicit systems. Numerical results are in good agreement with the published data and demonstrate the potential advantages of the high-order scheme in gaining both the high accuracy and the high efficiency. 展开更多
关键词 Discontinuous Galerkin scheme GMRES solver High order Implicit Runge–Kutta method Unsteady flows
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AN IMPLICIT SCHEME FOR THE NUMERICAL SOLUTION OF THE TIMEDEPENDENT HEAD/DISK DYNAMICS IN HARD DISK DRIVER BY FINITE ELEMENT METHOD 认领 引用
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作者 Hua Wei Tan Bing-lin He You-sheng 《Journal of Hydrodynamics》 SCIE EI 1994年第2期23-31,共9页
An implicit scheme discretized by the finite element method is developed for the numerical solution of the time-dependent head/disk dynamics. The scheme is second-order timeaccurate, which offers the flexibility of em... An implicit scheme discretized by the finite element method is developed for the numerical solution of the time-dependent head/disk dynamics. The scheme is second-order timeaccurate, which offers the flexibility of employing time steps compatible with the system dynamics while minimizing the numerical instability presented in most explicit methods. Good agreement is orreryed between the present calculations and D. K. Miu's work. 展开更多
关键词 implicit scheme finite element method time-dependent flying characteristics of slider
TRANSONIC FLOW CALCULATION OF EULER EQUATIONS BY IMPLICIT ITERATING SCHEME WITH FLUX SPLITTING 认领 引用
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作者 Liu Dao-zhi and Zha Ge-chengBeijing University of Aeronautics and Astronautics 《Chinese Journal of Aeronautics》 EI CAS 1991年第4期361-368,共8页
Three dimensional Euler equations are solved in the finite volume form with van Leer's flux vector splitting technique. Block matrix is inverted by Gauss-Seidel iteration in two dimensional plane while strongly im... Three dimensional Euler equations are solved in the finite volume form with van Leer's flux vector splitting technique. Block matrix is inverted by Gauss-Seidel iteration in two dimensional plane while strongly implicit alternating sweeping is implemented in the direction of the third dimension. Very rapid convergence rate is obtained with CFL number reaching the order of 100. The memory resources can be greatly saved too. It is verified that the reflection boundary condition can not be used with flux vector splitting since it will produce too large numerical dissipation. The computed flow fields agree well with experimental results. Only one or two grid points are there within the shock transition zone. 展开更多
关键词 TRANSONIC FLOW CALCULATION OF EULER EQUATIONS BY IMPLICIT ITERATING SCHEME WITH FLUX SPLITTING Flow
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Quinpi:Integrating Conservation Laws with CWENO Implicit Methods 认领 引用 被引量：1
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作者 G.Puppo M.Semplice G.Visconti 《Communications on Applied Mathematics and Computation》 2023年第1期343-369,共27页
Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Im... Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Implicit integration is quite straightforward for first-order schemes.High order schemes instead also need to control spurious oscillations,which requires limiting in space and time also in the linear case.We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor,which is used both to set up the nonlinear weights of a standard high order space reconstruction,and to achieve limiting in time.In this preliminary work,we concentrate on the case of a third-order scheme,based on diagonally implicit Runge Kutta(DIRK)integration in time and central weighted essentially non-oscillatory(CWENO)reconstruction in space.The numerical tests involve linear and nonlinear scalar conservation laws. 展开更多
关键词 Implicit schemes Essentially non-oscillatory schemes Finite volumes WENO and CWENO reconstructions
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A three dimensional implicit immersed boundary method with application 认领 引用 被引量：1
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作者 Jian Hao Luoding Zhu 《Theoretical & Applied Mechanics Letters》 EI CAS 2011年第6期22-25,共4页
Most algorithms of the immersed boundary method originated by Peskin are explicit when it comes to the computation of the elastic forces exerted by the immersed boundary to the fluid.A drawback of such an explicit app... Most algorithms of the immersed boundary method originated by Peskin are explicit when it comes to the computation of the elastic forces exerted by the immersed boundary to the fluid.A drawback of such an explicit approach is a severe restriction on the time step size for maintaining numerical stability.An implicit immersed boundary method in two dimensions using the lattice Boltzmann approach has been proposed.This paper reports an extension of the method to three dimensions and its application to simulation of a massive flexible sheet interacting with an incompressible viscous flow. 展开更多
关键词 immersed boundary method lattice-Boltzmann method implicit schemes fluid-structure-interaction bi-stability flag-in-wind
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An implicit method using contravariant velocity components and its application to calculations in a harbour-channel area 认领 引用 被引量：1
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作者 Shi Fengyan, Kong Yazhen and Ding Pingxing 《Acta Oceanologica Sinica》 SCIE CAS 1998年第4期423-432,共10页
The key problem in the computation of fluid dynamics using fine boundary-fitted grids is how to improve the numerical stability and decrease the calculating quantity. To solve this problem, implicit schemes should be ... The key problem in the computation of fluid dynamics using fine boundary-fitted grids is how to improve the numerical stability and decrease the calculating quantity. To solve this problem, implicit schemes should be adopted since explicit schemes may bring about a great increase in computation quantity according to the Courant-FrledrichsLewy condition. Whereas the adoption of implicit schemes is difficult to be realized because of the existence of two partial derivatives of surface elevations with respect to variables of alternative direction coordinates in each momentum equation in non-rectangular coordinates. With an aim to design an implicit scheme in non-reetangular ccordinates in the present paper, new momentum equations with the contravariant components of velocity vector are derived based on the shallow water dynamic equations in generalized curvilinear coordinates. In each equation, the coefficients before the two detivatives of surface elevations have different orders of magnitude, i. e., the derivative with the larger ceefficient rnay play a more important role than that with the smaller one. With this advantage, the ADI scheme can then be easily employed to improve the numerical stability and decrease the calculating quantity. The calculation in a harbour and a channel in Macau nearshore area shows that the implicit model is effective in calculating current fields in small size areas. 展开更多
关键词 Numerical model contravariant component of velocity vector implicit scheme
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Implicit discontinuous Galerkin method on agglomerated high-order grids for 3D simulations 认领 引用 被引量：3
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作者 Qin Wanglong Lyu Hongqiang +2 位作者 Wu Yizhao Zhou Shijie Chen Zhengwu 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2016年第6期1496-1505,共10页
High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discont... High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discontinuous Galerkin(DG) method for solving the 3D compressible Euler and Navier-Stokes equations. In this method, a fine linear mesh is first generated by standard commercial mesh generation tools. By taking advantage of an agglomeration method, a quadratic high-order mesh is quickly obtained, which is coarse but provides a high-quality geometry representation, thus very suitable for high-order computations. High-order discretizations are performed on the obtained grids with DG method and the discretized system is treated fully implicitly to obtain steady state solutions. Numerical experiments on several flow problems indicate that the agglomerated high-order mesh works well with DG method in dealing with flow problems of curved geometries. It is also found that with a fully implicit discretized system and a p-sequencing method, the DG method can achieve convergence state within several time steps which shows significant efficiency improvements compared to its explicit counterparts. 展开更多
关键词 Agglomeration Discontinuous Galerkin(DG) High-order Implicit scheme Navier-Stokes equations
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一类非线性延迟波动方程的线性隐式差分格式 认领 引用
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作者 谢建强 潘琬笛 汪灿 《阜阳师范大学学报（自然科学版）》 2026年第1期1-6,22,共6页
本文对非线性延迟波动方程建立了一个高效的二阶线性隐式差分格式。运用离散能量法,证明了差分格式在最大模意义下是无条件收敛的,其收敛阶为O(τ2+h2)。数值结果验证了所提格式的精度和有效性。
关键词 非线性延迟波动方程 降阶法 线性隐式差分格式 收敛性
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考虑转动与振动非平衡效应的双原子改进离散速度方法 认领 引用
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作者 韩林昌 李若愚 +2 位作者 杨鲤铭 李志辉 葛云龙 《空气动力学学报》 CSCD 北大核心 2026年第7期154-169,共16页
改进离散速度方法(improved discrete velocity method,IDVM)是一种多尺度模拟方法,可用于模拟从连续流到自由分子流的全流域流动。相比于传统离散速度方法(DVM),IDVM在保持其格式简洁性的同时,显著提升了近连续流区的计算精度与效率。... 改进离散速度方法(improved discrete velocity method,IDVM)是一种多尺度模拟方法,可用于模拟从连续流到自由分子流的全流域流动。相比于传统离散速度方法(DVM),IDVM在保持其格式简洁性的同时,显著提升了近连续流区的计算精度与效率。本文将全隐式迭代的IDVM推广至基于唯象论模型的Boltzmann方程,以实现考虑双原子分子转动、振动非平衡效应的跨流域流动模拟。在此基础上,将宏观控制方程的求解与Boltzmann模型方程的隐式离散相结合,在界面通量重构中引入碰撞效应,从而克服了传统DVM在连续流区计算效率低下的不足。通过对不同维度、马赫数下的跨流域问题进行测试,当前计算结果与同类数值方法及实验数据吻合良好。在连续和近连续流区,计算效率较传统半隐式DVM提升约一个数量级;引入振动自由度后,三维圆球绕流的阻力系数与实验值的相对误差由−1.19%改善至−0.67%。以上结果表明,该方法能够准确高效地模拟双原子分子非平衡流动问题。 展开更多
关键词 改进离散速度方法 转动非平衡 振动非平衡 唯象论模型 全隐式格式 跨流域流动
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Cahn-Hilliard方程的稳定化半隐格式切比雪夫谱方法 认领 引用
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作者 唐庶娟 罗贤兵 《数学杂志》 2026年第1期49-62,共14页
本文研究了Cahn-Hilliard方程的三种不同的数值格式:半隐式格式,一阶稳定化半隐格式,二阶稳定化半隐格式的问题.利用切比雪夫谱方法进行空间离散,有限差分法对时间离散的方法,得到Cahn-Hilliard方程的数值离散格式在数值实验中,验证了... 本文研究了Cahn-Hilliard方程的三种不同的数值格式:半隐式格式,一阶稳定化半隐格式,二阶稳定化半隐格式的问题.利用切比雪夫谱方法进行空间离散,有限差分法对时间离散的方法,得到Cahn-Hilliard方程的数值离散格式在数值实验中,验证了当数值解达到稳定时,对于不同的稳定化常数S,稳定格式所需时间步长相较于非稳定格式的1000倍.该方法验证了切比雪谱方法求解三种数值格式的有效性. 展开更多
关键词 Cahn-Hilliard方程 一阶稳定化半隐格式 二阶稳定化半隐格式 半隐格式 切比雪夫谱方法
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Efficient Difference Schemes for the Caputo-Tempered Fractional Diffusion Equations Based on Polynomial Interpolation 认领 引用
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作者 Le Zhao Can Li Fengqun Zhao 《Communications on Applied Mathematics and Computation》 EI 2021年第1期1-40,共40页
The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differen... The tempered fractional calculus has been successfully applied for depicting the time evolution of a system describing non-Markovian diffusion particles.The related governing equations are a series of partial differential equations with tempered fractional derivatives.Using the polynomial interpolation technique,in this paper,we present three efficient numerical formulas,namely the tempered L1 formula,the tempered L1-2 formula,and the tempered L2-1σformula,to approximate the Caputo-tempered fractional derivative of orderα∈(0,1).The truncation error of the tempered L1 formula is of order 2-α,and the tempered L1-2 formula and L2-1σformula are of order 3-α.As an application,we construct implicit schemes and implicit ADI schemes for one-dimensional and two-dimensional time-tempered fractional diffusion equations,respectively.Furthermore,the unconditional stability and convergence of two developed difference schemes with tempered L1 and L2-1σformulas are proved by the Fourier analysis method.Finally,we provide several numerical examples to demonstrate the correctness and effectiveness of the theoretical analysis. 展开更多
关键词 Caputo-tempered fractional derivative Polynomial interpolation Implicit ADI schemes Stability
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HIGH-ORDER COMPACT ADI SCHEMES FOR 2D SEMI-LINEAR REACTION-DIFFUSION EQUATIONS WITH PIECEWISE CONTINUOUS ARGUMENT IN REACTION TERM 认领 引用
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作者 Bo Hou Chengjian Zhang 《Journal of Computational Mathematics》 SCIE CSCD 2026年第1期191-212,共22页
This paper deals with the numerical solutions of two-dimensional(2D)semi-linear reaction-diffusion equations(SLRDEs)with piecewise continuous argument(PCA)in reaction term.A high-order compact difference method called... This paper deals with the numerical solutions of two-dimensional(2D)semi-linear reaction-diffusion equations(SLRDEs)with piecewise continuous argument(PCA)in reaction term.A high-order compact difference method called Ⅰ-type basic scheme is developed for solving the equations and it is proved under the suitable conditions that this method has the computational accuracy O(τ2+hx4+hy4),where τ,h_(x )and hy are the calculation stepsizes of the method in t-,x-and y-direction,respectively.With the above method and Newton linearized technique,a Ⅱ-type basic scheme is also suggested.Based on the both basic schemes,the corresponding Ⅰ-and Ⅱ-type alternating direction implicit(ADI)schemes are derived.Finally,with a series of numerical experiments,the computational accuracy and efficiency of the four numerical schemes are further illustrated. 展开更多
关键词 Semi-linear reaction-diffusion equations Piecewise continuous argument Highorder compact difference methods Alternating direction implicit schemes Computational accuracy and efficiency
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