# 微分方程数值解速通

微分方程## Initial Value Problems

Problem formulation:

### Numerical Differentiation Based Methods

- Grids
- Evaluate the differential equation at the grid points
- Replace the derivative by a difference quotient
where is the truncation error. The equation becomes - Drop the truncation error term
And the initial condition gives - Error analysis of
. The method is first order accurate.

### Stability

Use the model problem

- A-stable:
, is bounded? - L-stable: For all
, , is bounded? - Zero-stable: Let
, , is bounded?

### Numerical Integration Based Methods

- Grids
- Integrate the differential equation over the interval
That is - Replace the integral by a quadrature formula
- Drop the error term

Other notes:

- The idea of Runge-Kutta methods is to replace future
values in implicit methods by values predicted by an explicit method. - Extension and ghost points are used to handle boundary conditions, by adding extra points to the grid.
- Conservative methods are used to preserve the total energy of the system.

## Two-Point Boundary Value Problems

Problem formulation:

Using the five-step finite difference method, we will eventually get a system of linear equations to solve. Constant boundary conditions will give a tridiagonal system:

where

The tridiagonal matrix can be solved in

- Theorem 1:
is invertible when either is strictly diagonally dominant, or is duagonally dominant, and , .

- Theroem 2: If
are continuous on , then the solution of the BVP exists and is unique when the step size satisfies - Theorem 3 (Error Analysis): When the conditions of Theorem 2 are satisfied, the error of the finite difference method is
where is a constant ( , ), is the truncation error, and are the boundary errors.

## Diffusion Problems

Problem formulation: The inhomogeneous heat equation

with initial condition

5-step explicit finite difference method will result in a matrix form of

where

where

and deduce

- A-Stability: The method is A-stable if
for all . - L-Stability: The method is L-stable if
as .

## Advection Problems

Problem formulation: The advection equation

with initial condition

CFL condition: The numerical domain of dependence must contain the physical domain of dependence.

## Numerical Wave Propagation

Problem formulation: The wave equation

with zero boundary conditions

Solution properties: Substitute the plane wave solution

into the wave equation, and extract the coefficients of

and the group velocity,

- Dispersive:
depends on . - Dissipative:
.

## Elliptic Problems

Problem formulation: The Laplace equation

with given boundary conditions.

Key points:

- Banded matrices can be used to solve the Laplace equation.
- Positive definite matrices (Geršgorin circle theorem)
- Iterative methods