# 微分方程数值解作业 5

微分方程## Problem 1

Question | Dispersion Relation | Phase Velocity | Group Velocity | Dispersive | Dissipative |
---|---|---|---|---|---|

(a) | Yes | No | |||

(b) | Yes | Yes | |||

(c) | Yes | No |

### Question (a)

With the plane wave solution

we have

That simplifies to

which holds for all

and this is the dispersion relation. The phase velocity

The group velocity

The equation is dispersive since

### Question (b)

With the plane wave solution, we have

That simplifies to

Therefore the dispersion relation is

Since

The phase velocity

The group velocity

It can be shown that

Also,

which is true since

The equation is dispersive since

### Question (c)

With the plane wave solution, we have

which simplifies to

The dispersion relation is

The phase velocity

The group velocity

The equation is dispersive since

## Problem 2

### Question (a)

### Question (b)

From (5.31) we have

With the initial condition in

That is

All desired

And then compute

### Question (c)

The method satisfies CFL condition, since the system described by

### Question (d)

From (5.31) we have

With the initial condition in

Therefore

The stencil for

The CFL condition is the same as in the textbook.

## Problem 3

### Question (a)

- Use a uniform grid
where and - Evaluate the difference equation at
- Using centered differences to approximate the derivatives we have
This can be rearranged to where is the truncation error. - Drop the error term to get the finite difference approximation
for and . The first boundary condition in (5.2) can be approximated by The second boundary condition in (5.3) can be handled by introducing ghost points at , described in . With we can calculate for

The stencil for this method is

and

at the first time step The numerical domain of dependence consists of the points