Numerical Simulations of Partial Differential Equations: Time‐dependent Problems
Thierry Goudon · 2016
This chapter considers the numerical approximation of the heat equation in one spatial dimension. When the problem spans all space (x ∈ R), it can be simply resolved by the Fourier transform. The simplest scheme depends upon an explicit Euler-type discretization for the time derivative and a finite difference discretization. The numerical analysis of this equation combines both the difficulties of the stationary problems (the choice of infinite-dimensional norms) and the stability issues related to temporal evolution. In order to study von Neumann stability, the chapter limits the problem to one with periodic condition: it considers heat equation applied in R with the condition u (t, x) = u (t, x + 2π). Finite volume methods are especially suited for dealing with conservation laws because these equations correspond exactly to balances (of mass, energy, etc.), where the gains and losses in a domain arise from exchanges at the interfaces of the domain.