Main types

• Finite element methods for complex geometries
• Finite volume methods for Hyperbolic geometry
• Finite difference method for simple geometry.

Now have time and space.

Simplest approach is again finite difference discretization. Now discretizing time and space.

Numerical stability

von Neumann analysis or discrete Fourier analysis. Plug imaginary (oscillatory) exponential into the finite difference formula, and see if some mode blows up (by the amplification factor being greater than 1), or not. Define region of stability thus.

PDEs can also be stiff for same reasons as ODEs, and then need to use implicit methods too. A non-linear example is the Kuramoto-Sivashinsky equation.

Order of accuracy

Defined now for both timestep $k$ and space step $h$ (see notes). To improve order of accuracy over straightforward Euler method (which is first order in $k$) we use the trapezoidal rule, which is symmetric in $t$ (so that first order errors cancel, and is thus 2nd order in $k$). In case of heat equation it's known as Crank-Nicolson formula. In the case of heat equation, it's known as the leap frog formula (1928).

Reaction-diffusion equations and other stiff PDEs. Can use exponential integrator methods... Solitons

Finite differencing in general grids

Not necessarily equally-spaced.

Principle:

1. At each $x_j$ decide which data, from neighbouring points, $v_{j-r}, ..., v_{j+s}$ to use.

2. Interpolate these data by polynomial of degree $r+s$.

3. Finite difference approximation to $k$th derivative is: $p^{(k)}(x_j)$.

We don't do these steps explicitly at every step, rather there are slick algorithms to get a formula for general $v$s for arbitrary grids $x_j$s, and one uses that formula. See B. Fornberg, “Generation of finite difference formulas on arbitrarily spaced grids,” Math. Comput. 51 (1988), 699-706 and B. Fornberg, “Calculation of weights in finite difference formulas”, SIAM Review 40 (1998), 685-691.

In multiple space dimension same principles apply, but the system of equations needed to be solved for implicit methods corresponds to a matrix that has a much wider "band" (i.e. set of non-zero diagonals) than for 1 dimension. The structure of this matrix, in the case of discretizing the Laplacian is the famous "discrete or lattice Laplacian" (related to the Graph laplacian). See notes. This Laplacian can often be written as a Kronecker sum.

Spectral methods

Examples of Differential Equations, with nice explanations:

Trefethen et al.'s PDE COFFEE TABLE BOOK

Reaction-diffusion equations in Morphogenesis