1 Difference methods for the Fisher equation

1.1 Introduction

The Fisher equation belongs to the class of reaction-diffusion equations. In fact, it is one of the simplest semilinear reaction-diffusion equations, the one which has the inhomogeneous term f(u)=λu(1−u), which can exhibit traveling wave solutions that switch between equilibrium states given by f(u)=0. Such an equation occurs, e. g., in ecology, physiology, combustion, crystallization, plasma physics and in general, phase transition problems. Fisher proposed this equation in 1937 to describe the spatial spread of an advantageous allele and explored its traveling wave solutions [12]. In the same year (1937) as Fisher, Kolmogorov, Petrovskii and Piskunov introduced a more general reaction-diffusion equation [18]. In this chapter, we consider the following initial and boundary value problem of a one-dimensional Fisher equation:

where λ is a positive constant, functions φ(x), α(t), β(t) are all given and φ(0)=α(0), φ(L)=β(0). Suppose that the problem (1.1)–(1.3) has a smooth solution.

Before introducing the difference scheme, a priori estimate on the solution of the problem (1.1)–(1.3) is given.

1.2 Notation and lemmas

In order to derive the difference scheme, we first divide the domain [0,L]×[0,T]. Take two positive integers m, n. Divide [0,L] into m equal subintervals, and [0,T] into n subintervals. Denote h=L/m, τ=T/n; xi=ih, 0⩽i⩽m; tk=kτ, 0⩽k⩽n; Ωh={xi∣0⩽i⩽m}, Ωτ={tk∣0⩽k⩽n};Ωhτ=Ωh×Ωτ. We call all of the nodes {(xi,tk)∣0⩽i⩽m} on the line t=tk the k-th time-level nodes. In addition, denote xi+12=12(xi+xi+1), tk+12=12(tk+tk+1), r=τh2.

Denote

For any grid function u∈Uh, introduce the following notation:

It follows easily that

Suppose u,v∈Uh. Introduce the inner products, norms and seminorms as

If Uh is a complex space, then the corresponding inner product is defined by

with v¯i the conjugate of vi.

Denote

For any w∈Sτ, introduce the following notation:

It is easy to know that

Suppose u={uik∣0⩽i⩽m,0⩽k⩽n} is a grid function defined on Ωhτ, then v={uik∣0⩽i⩽m} is a grid function defined on Ωh, w={uik∣0⩽k⩽n} is a grid function defined on Ωτ.