Finite Differences and the Method of Lines

Abstract

In this chapter, finite difference (FD) approximations and the method of lines, which combine FD with available time integrators, are discussed. First, a convection diffusion-reaction PDE is used to introduce a few basic FD schemes and addresses the concept of stability of the numerical scheme. As a rule of thumb, centered FD schemes appear as good choices for approximating second-order (diffusion) operators, whereas upwind FD schemes are preferable to first-order (convection) operators. Once stability is ensured, accuracy can be adjusted by selecting appropriately the fineness of the spatial grid and the order of the FD approximation (i.e. the stencil of points on which the FD approximation is built). The calculation of FD approximation can be conveniently organized using differentiation matrices, which are easy to manipulate in MATLAB, SCILAB, or OCTAVE. FD can also be efficiently computed on arbitrarily spaced grids using an algorithm due to Fornberg. This chapter continues with a presentation of different methods to take the boundary conditions into account. Boundary conditions are an essential part of the IBVP definition, and several methods for translating the BCs in the code implementation are presented. Finally, attention is paid to the computation of the Jacobian matrix of the ODE system, which is used by various solvers.