蔣沁紗　陳浩

四川師范大學(xué)學(xué)報(bào)（自然科學(xué)版）第46卷第5期

摘要：考慮一類空間變系數(shù)反應(yīng)擴(kuò)散方程的快速算法.針對二階改進(jìn)道格拉斯分裂時(shí)間離散所得線性代數(shù)系統(tǒng)，構(gòu)造一類雙參數(shù)交替分裂迭代方法.分析格式的收斂性，給出最優(yōu)參數(shù)的取值，并獲得相應(yīng)預(yù)處理子.數(shù)值結(jié)果驗(yàn)證新方法的有效性及相比單參數(shù)分裂迭代格式的優(yōu)越性.

關(guān)鍵詞：變系數(shù)反應(yīng)擴(kuò)散方程; 改進(jìn)道格拉斯分裂方法; 雙參數(shù); 交替分裂迭代方法; 預(yù)處理子

中圖分類號：O241.82; O241.6 文獻(xiàn)標(biāo)志碼：A 文章編號：1001-8395（2023）05-0638-08

1離散

2交替分裂迭代算法

3數(shù)值實(shí)驗(yàn)

4結(jié)束語

本文考慮了變系數(shù)反應(yīng)擴(kuò)散方程的快速算法，針對改進(jìn)道格拉斯分裂時(shí)間離散所得的線性代數(shù)系統(tǒng)，構(gòu)造了一類雙參數(shù)交替分裂迭代法，分析了其收斂性及最優(yōu)參數(shù)的取值.同時(shí)，將其與GMRES結(jié)合，構(gòu)造了一類預(yù)處理GMRES的方法，數(shù)值結(jié)果驗(yàn)證了新方法的收斂性.

參考文獻(xiàn)

[1] HUNDSDORFER W， VERWER J. Advection-diffusion Discretizations[M]. Berlin：Springer，2003：215-323.

[2] 孫志忠. 非線性發(fā)展方程的有限差分方法[M]. 北京：科學(xué)出版社，2018.

[3] ZHOU Z G， LIANG D. Mass-preserving time second-order explicit-implicit domain decomposition schemes for solving parabolic equations with variable coefficients[J]. Computational and Applied Mathematics，2018，37（4）：4423-4442.

[4] EVANS L C. Partial Differential Equations[M]. Providence：American Mathematical Society，1999.

[5] HESTHAVEN J， GOTTLIEB S， GOTTLIEB D. Spectral Methods for Time-dependent Problems[M]. Cambridge：Cambridge University Press，2007.

[6] ISERLES A. A First Course in the Numerical Analysis of Differential Equations[M]. Cambridge：Cambridge University Press，1996.

[7] ARRARS A， INHOUT K J， HUNDSDORFER W， et al. Modified Douglas splitting methods for reaction-diffusion equations[J]. BIT Numerical Mathematics，2017，57（2）：261-285.

[8] PEACEMAN D W， RACHFORD H H Jr. The numerical solution of parabolic and elliptic differential equations[J]. Journal of the Society for Industrial and Applied Mathematics，1955，3（1）：28-41.

[9] DOUGLAS J. Alternating direction methods for three space variables[J]. Numerische Mathematik，1962，4（1）：41-63.

[10] BAI Z Z， GOLUB G H， NG M K. Hermitian and Skew-Hermitian splitting methods for non-Hermitian positive definite linear systems[J]. SIAM Journal on Matrix Analysis and Applications，2003，24（3）：603-626.

[11] CHEN H. A splitting preconditioner for the iterative solution of implicit Runge-Kutta and boundary value methods[J]. BIT Numerical Mathematics，2014，54（3）：607-621.

[12] CHEN H. Generalized Kronecker product splitting iteration for the solution of implicit Runge-Kutta and boundary value methods[J]. Numerical Linear Algebra With Applications，2015，22（2）：357-370.

[13] CHEN H， L W， ZHANG T T. A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations[J]. Journal of Computational Physics，2018，360：1-14.

[14] BAI Z Z， LU K Y， PAN J Y. Diagonal and Toeplitz splitting iteration methods for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations[J]. Numerical Linear Algebra With Applications，2017，24（4）：e2093.

[15] LIN X L， NG M K， SUN H W. A splitting preconditioner for Toeplitz-like linear systems arising from fractional diffusion equations[J]. SIAM Journal on Matrix Analysis and Applications，2017，38（4）：1580-1614.

[16] 蔣沁紗，陳浩. 空間變系數(shù)反應(yīng)擴(kuò)散方程的一類交替分裂預(yù)處理迭代方法[J]. 重慶師范大學(xué)學(xué)報(bào)（自然科學(xué)版），2022，39（5）：83-90.

[17] HORN R A， JOHNSON C R. Topics in Matrix Analysis[M]. Cambridge：Cambridge University Press，1991.

[18] SAAD Y. Iterative Methods for Sparse Linear Systems[M]. 2nd ed. Philadelphia：SIAM，2003.

A Class of Alternating Splitting Preconditioning Method with Two Parameters

for Reaction-Diffusion Equations with Variable Coefficients in SpaceJIANG Qinsha，CHEN Hao（School of Mathematical Sciences， Chongqing Normal University， Chongqing 401331）

Abstract：This paper consider fast algorithms for solving a class of reaction-diffusion equations with variable coefficients. We propose an alternating splitting iterative method with two parameters for solving the linear algebraic systems resulting from the modified Douglas splitting discretization of the reaction-diffusion equations. We show that the proposed scheme is convergent and the optimal parameters are given. A splitting preconditioner is also derived for the linear system. Numerical results show that the proposed methods is effective and superior to the splitting iterative scheme with a single parameter.

Keywords：reaction-diffusion equation with variable coefficients; modified Douglas splitting method; two parameters; alternating splitting iteration method; preconditioner

2020 MSC：65F10; 65L06; 65N22

（編輯 余毅）