非线性反应扩散方程的数值解

Numerical Solution of a Nonlinear Reaction-Diffusion Equation

摘要: 本文采用Petrov-Galerkin有限元法构造了非线性反应扩散方程的数值格式,既适用于全场性的初值,也适用于局部性的初值,利用全场性初值求得的进波数值解与精确解高度吻合,证明本方法与其它数值方法比较,有更高的精度和稳定性,利用各种局部性初值所算出的数值解表明,任何局部的扰动都会得到充分的发展,且当时间充分长后,演变为向左、右传播的行波,其波前的形状及传播速度完全由系统本身所决定,而与初值的类型无关.
- 反应扩散方程 /
- Petrov-Galerkin有限元法 /
- 行波
Abstract: A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exact and numerical solutions of progressive waves shows that this numerical scheme is quite accurate, stable andefflcient. It is also shown that any local disturbance will spread, have a full growth and finally form two progressive waves propagating in both directions. The shape and the speed of the long term progressive waves are determined by the system itself, and do not depend on the details of the initial values.
- reaction-diffusion equation /
- Petrov-Galerkin finite element method /
- progressive wave

[1]	Fisher,R.A.,The wave of advance of advantageous genes,Ann.of Eugen,7(1936),355-369.
[2]	Canosa,J.,Diffusion in nonlinear multiplication media.J.Math.Phys.,10(1969),1862-1868.
[3]	Canosa.J.,On a nonlinear diffusion equation describing population,growth,IBM J.Res.Develop.,17(1973),307-313.
[4]	Kolmogoroff,A.,I.Petrovaky and N.Piscounoff.Study of the diffusion equation with growth of the quantity of matter and its application to biology problem,Bulletin de I'Université délatã Moscou.Séreinternationale,Section A,1(1937).
[5]	Ramos,J.I.,Numerical solution of reaction-diffusion equation,Int.J.Comp.Maths.18(1985).43-66. 141-162;289-310.
[6]	Griffiths,D.F.and A.R.Mitchell.Stable periodic bifurcations of an explicit discretization of a nonlinear partial differential equation in reaction diffusion,IMA J.of Numer.Analy.,8(1988) 435-454.
[7]	Sachdev,P.L.,Nonlinear Diffusive Waves,Cambridge University Press(1987).103-105.
[8]	Wang,X.Y.,Exact and explicit solitary wave solutions for the generalized Fisher equation,Physics Letter A,131,(1988),277-279.
[9]	Gazdag,J.and J.Canosa,Numerical solution of Fisher's equation,J.Appl.Probab.,11(1974),445-457.
[10]	Evans,D.J.and M.S.Sahimi,The alternating group explicit(AGE)iterative method to solve parabolic and hyperbolic partial differential equations,Ann.of Numerical Fluid Mechanics and Heat Transfer,2(1989),283-389.
[11]	Tang,S.and R.O.Weber,Numerical study of Fisher's equation by a Petrov-Galerkin finite element method,J.Austral.Math.Soc.,Ser.B.(in press)