Finite Difference Methods for Partial Differential Equations
In this chapter, we initially give an introduction to methods for computing derivatives and partial derivatives using discrete differential operators and discuss the connection to Taylor series.
The chapter moves on to the topic of solving PDEs using finite difference methods. We discuss implicit and explicit methods and boundary conditions. The chapter also covers the categories of PDEs: elliptic, hyperbolic and parabolic as well as the important notions of consistence, convergence and stability. Finally, there is a discussion of 2D and 3D problems and problems on irregular grids.
At the end of the chapter, linear interpolation is discussed. We discuss both linear interpolation on a regular 2D or 3D grid and linear interpolation in a simplicial complex using barycentric coordinates.
As in the previous two chapters, this chapter is intended as a brush up and a point of reference. The reader who wishes to know more is referred to the literature.
KeywordsFinite Difference Method Finite Difference Scheme Implicit Method Finite Difference Equation Irregular Grid
- 3.Osher, S.J., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces, 1st edn. Springer, Berlin (2002) Google Scholar