热传导方程Neumann边界值问题的紧差分格式
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摘要
W．Y．Liao和J．P．Zhu(J．Comp．Math．，(2005))用Pade逼近方法对下列热传导方程Neumann边界值问题
(?)w/(?)t=(?)~2w/(?)x~2+f(x，t)，0＜x＜1，0＜t≤T，
(?)w/(?)x(0，t)=α(t)，(?)w/(?)x(1，t)=β(t)，0＜t≤T，
w(x，0)=w_0(x)，0≤x≤1。建立了一个具有高阶逼近精度的差分格式，但未见相关的理论分析。本文证明了这一格式是无条件稳定且是收敛的，其收敛阶为O(Υ~2+h~(3.5))。此外对此方法进行了改进，降低了边界处离散产生的误差，建立了一个精度更高的差分格式，证明了其无条件稳定性和收敛性，收敛阶为O(Υ~2+h~4)。数值计算的结果验证了理论结果，
本文第二部分利用Keller Box格式及降阶法技术，对一维热传导方程的Neumann边界值问题进行了研究，建立了一个高阶差分格式，同时分析了该差分格式解的存在唯一性、收敛性和稳定性，并用数值算例对理论结果进行了验证。
W. Y. Liao and J. P. Zhu (J. Comp. Math.,(2005)) derives a high order difference scheme for the following one-dimensional heat equation.
but there is no theoretical analysis. In this paper, we prove that the scheme is unconditionally stable, and is convergent with the convergence order of O(τ~2 + h~(3.5)). Moreover, we improve the scheme and derive a more accurate difference scheme by reducing the error produced by discretization at the boundaries. We also prove that it is absolutely stable and convergent with the convergence order of O(τ~2 + h~4). Numerical examples testify the theoretical results.
In the second part, we study the one-dimensional heat equation with Neumann boundary conditions by the method of reduction of order and Keller Box scheme. By introducing a new variable v = w_x, we eliminate the error produced by the discretization for the derivative on boundaries, derive a fourth-order difference scheme, and analyze existence, uniqueness, convergence and stability of difference solution. A numerical example demonstrates the theoretical results.
(?)w/(?)t=(?)~2w/(?)x~2+f(x，t)，0＜x＜1，0＜t≤T，
(?)w/(?)x(0，t)=α(t)，(?)w/(?)x(1，t)=β(t)，0＜t≤T，
w(x，0)=w_0(x)，0≤x≤1。建立了一个具有高阶逼近精度的差分格式，但未见相关的理论分析。本文证明了这一格式是无条件稳定且是收敛的，其收敛阶为O(Υ~2+h~(3.5))。此外对此方法进行了改进，降低了边界处离散产生的误差，建立了一个精度更高的差分格式，证明了其无条件稳定性和收敛性，收敛阶为O(Υ~2+h~4)。数值计算的结果验证了理论结果，
本文第二部分利用Keller Box格式及降阶法技术，对一维热传导方程的Neumann边界值问题进行了研究，建立了一个高阶差分格式，同时分析了该差分格式解的存在唯一性、收敛性和稳定性，并用数值算例对理论结果进行了验证。
W. Y. Liao and J. P. Zhu (J. Comp. Math.,(2005)) derives a high order difference scheme for the following one-dimensional heat equation.
but there is no theoretical analysis. In this paper, we prove that the scheme is unconditionally stable, and is convergent with the convergence order of O(τ~2 + h~(3.5)). Moreover, we improve the scheme and derive a more accurate difference scheme by reducing the error produced by discretization at the boundaries. We also prove that it is absolutely stable and convergent with the convergence order of O(τ~2 + h~4). Numerical examples testify the theoretical results.
In the second part, we study the one-dimensional heat equation with Neumann boundary conditions by the method of reduction of order and Keller Box scheme. By introducing a new variable v = w_x, we eliminate the error produced by the discretization for the derivative on boundaries, derive a fourth-order difference scheme, and analyze existence, uniqueness, convergence and stability of difference solution. A numerical example demonstrates the theoretical results.
引文
[1] B. Courbet, J. P. Croisille. Finite volume box schemes on triangular meshes, Math. Model. Numer. Anal, 32(5)(1998), 631-649.
[2] B. Courbet, Two-point schemes for numerical simulation of flows, La Recherche Aerospatiale
[3] J. P. Croisille, Keller's box-scheme for the one-dimensional stationary convec.tion-diffusion equation, Computing, 68(2002), 37-63.
[4] J. P. Croisille, I. Greff, An efficient box-scheme for convection-diffusion equations with sharp contrast in the diffusion coefficients, Computers & Fluids 34(4-5)(2005), 461-489.
[5] Y. Gu, W. Y. Liao, J. P. Zhu, An efficient high order algorithm for solving systems of 3-D reactiondiffusion equations, J. Comp. Appl. Math. 155(2003), 1-17.
[6] H. B. Keller, A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, ed.), New York Academic Press(1971), 327-350.
[7] W. Y. Liao, J. P. Zhu, and Q. M. Khaliq, An efficient high order algorithm for solving systems of reaction-diffusion equations, Numer. Methods Partial Differential Eq 18(2002), 340-354.
[8] W. Y. Liao, J. P. Zhu, Abdul Q.M. Khaliq, A fourth-order compact algorithm for nonlinear reaction-diffusion equations with Neumann boundary conditions, Numer. Methods Partial Differential Eq., 22(2006).
[9] Z. Z. Sun, A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. Comp. App. Math., 76(1-2)(1996), 137-146.
[10] Z. Z. Sun, The method of the reduction of order for the numerical solution to elliptic differential equations, J. Southeast Univ., 23(6)(1996), 8-16.
[11] Z. Z. Sun, A higher Order Accurate Seheme for Heat Equation in Unbounded Domains Using Artificial Boundary Conditions. (To be published)
[12] Z. Z. Sun, A new class of difference schemes for linear parabolic differential equations, Numer. Math. Sinca, 16(2)(1994), 115-130.
[2] B. Courbet, Two-point schemes for numerical simulation of flows, La Recherche Aerospatiale
[3] J. P. Croisille, Keller's box-scheme for the one-dimensional stationary convec.tion-diffusion equation, Computing, 68(2002), 37-63.
[4] J. P. Croisille, I. Greff, An efficient box-scheme for convection-diffusion equations with sharp contrast in the diffusion coefficients, Computers & Fluids 34(4-5)(2005), 461-489.
[5] Y. Gu, W. Y. Liao, J. P. Zhu, An efficient high order algorithm for solving systems of 3-D reactiondiffusion equations, J. Comp. Appl. Math. 155(2003), 1-17.
[6] H. B. Keller, A new difference scheme for parabolic problems, Numerical solutions of partial differential equations, ed.), New York Academic Press(1971), 327-350.
[7] W. Y. Liao, J. P. Zhu, and Q. M. Khaliq, An efficient high order algorithm for solving systems of reaction-diffusion equations, Numer. Methods Partial Differential Eq 18(2002), 340-354.
[8] W. Y. Liao, J. P. Zhu, Abdul Q.M. Khaliq, A fourth-order compact algorithm for nonlinear reaction-diffusion equations with Neumann boundary conditions, Numer. Methods Partial Differential Eq., 22(2006).
[9] Z. Z. Sun, A second-order accurate finite difference scheme for a class of nonlocal parabolic equations with natural boundary conditions, J. Comp. App. Math., 76(1-2)(1996), 137-146.
[10] Z. Z. Sun, The method of the reduction of order for the numerical solution to elliptic differential equations, J. Southeast Univ., 23(6)(1996), 8-16.
[11] Z. Z. Sun, A higher Order Accurate Seheme for Heat Equation in Unbounded Domains Using Artificial Boundary Conditions. (To be published)
[12] Z. Z. Sun, A new class of difference schemes for linear parabolic differential equations, Numer. Math. Sinca, 16(2)(1994), 115-130.