1. Show that the following scheme is consistent with the one-way wave equation ut + aux =
f (x, t):
a un+1 un+1 un un 1
un+1 un
m
m+1
m
m
m
n
+
+m
= fm .
t
2
x
x
2. Show that the following scheme is c
4
Finite Volume Methods
In this chapter we begin to study nite volume methods for the solution of conservation
laws and hyperbolic systems. The fundamental concepts will be introduced, and then we
wil
3
Characteristics and Riemann Problems for Linear
Hyperbolic Equations
In this chapter we will further explore the characteristic structure of linear hyperbolic systems of equations. In particular, we
1. Verify that the Lax-Friedrichs ux is consistent including Lipschitz continuity.
2. Show that the Lax-Wendro method on ut + aux = 0 can be derived by approximating
n
n
n
u(xj at, tn ) using quadrati
ul , x < st
where
ur , x > st
s = (ul + ur )/2 with ul > ur is a weak solution to Burgers equation by showing that
1. Consider the Burgers equation ut + f (u)x = 0. Verify that u(x, t) =
+
+
[t u + x
1. Consider the following equation
ut = uxx + uyy , t > 0, (x, y ) (0, 1) (0, 1)
u(0, y, t) = u(1, y, t) = 0, y [0, 1], t 0
u(x, 0, t) = u(x, 1, t) = 0, x [0, 1], t 0
u(x, y, 0) = sin x sin 2y, (x, y
1. Write a code to solve
ut = uxx + uyy , t > 0, (x, y ) (0, 1) (0, 1)
u(0, y, t) = u(1, y, t) = 0, y [0, 1], t 0
u(x, 0, t) = u(x, 1, t) = 0, x [0, 1], t 0
u(x, y, 0) = sin x sin 2y, (x, y ) [0, 1] [
58
Parabolic equations in one space variable
[Observe that for this model problem the largest error is always
in the rst time step.]
2.3
Suppose that the mesh points xj are chosen to satisfy
0 = x0 <
1. [This is HW1.pdf #3 and #4] Consider the following dierential equation
ut = uxx , t > 0, 0 x 1.
u(0, t) = 0 and u(1, t) = 0
u(x, 0) = sin 4x
and try the following scheme to solve the equation
u n 2
1. Analyze the stability and convergence of the following dierence schemes.
(a) un+1 = Run1 + (1 R)un where R =
k
k
k
(b) Where r =
at
x .
t
,
x2
un+1 = un + r
k
k
1n
4
5
4
1
u
.
+ un un + un un
12
1. Show that the schemes of the form
n
n
n
vm+1 = vm+1 + vm1
are stable if | + | | is less than or equal to 1. Conclude that the Lax-Friedrichs scheme
n
n
n
n
vm+1 1 (vm+1 + vm1 )
v n vm1
2
+ a m+1
=0
6
High-Resolution Methods
In Chapter 4 we developed the basic ideas of Godunovs method, an upwind nite volume method for hyperbolic systems, in the context of constant-coefcient linear systems.
Goduno