Finite Difference Methods for Parabolic Equations

Abstract

(1)  ut −∆u = f in Ω× (0, T ), u = 0 on ∂Ω× (0, T ), u(·, 0) = u0 in Ω. Here u = u(x, t) is a function of spatial variable x ∈ Ω ⊂ R and time variable t ∈ (0, T ). The ending time T could be +∞. The Laplace operator ∆ is taking with respect to the spatial variable. For the simplicity of exposition, we consider only homogenous Dirichlet boundary condition and comment on the adaptation to Neumann and other type of boundary conditions. Besides the boundary condition on ∂Ω, we also need to assign the function value at time t = 0 which is called initial condition. For parabolic equations, the boundary ∂Ω× (0, T )∪Ω×{t = 0} is called the parabolic boundary. Therefore the initial condition can be also thought as a boundary condition.