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 consistent with the equation ut + cutx + aux = f :
u n u
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
will focus on rst-order accurate methods for linear equati
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 will study solutions to the Riemann problem, which is
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 quadratic interpolation based on the points Uj 1 , Uj and Uj +1
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 f (u)]dxdt] =
0
(x, 0)u(x, 0)dx
1
is satised for all C
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 ) [0, 1] [0, 1].
Code with (i) Crank-Nicolson scheme, (
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] [0, 1].
Observation ?
2. Discuss the stability of the fo
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 < x1 < x2 < < xJ 1 < xJ = 1
but are otherwise arbitrary.
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 2u n + u n 1
un+1 un
m
m
m
m
= m+1
.
t
x2
How does the s
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 k2 3 k1 2 k 3 k+1 12 k+2
2. Show that the following die
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
t
2x
is stable if |a| is less than equal to 1, where =
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.
Godunovs method is only rst-order accurate and introduces a g