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