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BRADLEY J. LUCIER""Department of Mathematics, Purdue University,
West Lafayette, Indiana 47907.
The work of the first author was not supported by the
Wolf Foundation.
and
DOUGLAS N. ARNOLD""Department of Mathematics, University
of Maryland, College Park, Maryland 20742.
Abstract:
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The AmS-TeX SIAM style file (amstexsiam.sty), which is a modification of
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1. Introduction
We are concerned with numerical approximations to the so-called
porous-medium equation [#!6!#],
We assume that the initial data u0(x) has bounded support, that
0≤u0≤M, and that
φ(u0)x∈.
It is well known that a unique solution u(x, t) of (1.1) exists,
and that u satisfies
0≤
u≤
M and
φ(
u( ⋅ ,
t))
x≤
φ(
u0)
x.
1.2
If the data has slightly more regularity, then this too is satisfied
by the solution. Specifically, if m is no greater than two and
u0 is Lipschitz continuous, then
u( ⋅ , t) is also Lipschitz;
if m is greater than two and
(u0m-1)x∈, then
(u( ⋅ , t)m-1)x∈
(see [3]). (This will follow from results presented here, also.)
We also use the fact that the solution u is Hölder continuous in t.
2. error bounds
After a simple definition, we state a theorem
that expresses the error of approximations uh in
terms of the weak truncation error E.
Definition 2.1A definition
is the same as a theorem set in roman
type.
Theorem 2.1
Let {uh} be a family of approximate solutions satisfying
the following conditions for
0≤t≤T:
For all x∈ and positive t,
0≤uh(x, t)≤M;
Both u and uh are Hölder–α in x
for some
α∈(0, 1∧1/(m - 1)); uh is right continuous in t;
and uh is Hölder continuous in t on
strips
×(tn, tn+1), with the set {tn} having no
limit points; and
There exists a positive function
ω(h, ε) such that:
whenever
{wε}0 < ε≤ε0 is a family of functions
in X for which
"(a)" there is a sequence of positive numbers ε tending
to zero, such that for these values of
ε,
| wε|∞≤1/ε,
"(b)" for all positive
ε,
| wxε(, t)|≤1/ε2, and
"(c)" for all
ε > 0,
where p is some number not exceeding 1,
then
| E(uh, wε, T)|≤ω(h, ε).
Then, there is a constant
C = C(m, M, T) such that
|
u -
uh|
∞,×[0, T]≤
Csup
(
u0(
x) -
uh(
x, 0))
w(
x, 0) d
x +
ω(
h,
ε) +
εα,
2.1
where the supremum is taken over all
w∈X.
ProofLet z be in X. Because
E(u,,)≡ 0,
Equation (1.5) implies that
Δuz|
T0dx =
Δu(
zt +
φ[
u,
uh]
zxx) d
x d
t -
E(
uh,
z,
t),
2.2
where
Δu = u - uh and
φ[
u,
uh] =
.
Extend
φ[u, uh](⋅, t) = φ[u, uh](⋅, 0) for negative t, and
φ[u, uh](⋅, t) = φ[u, uh](⋅, T)
for t > T.Fix a point x0 and a number
ε > 0. Let
jε
be a smooth function of x with integral 1 and support in
[- ε, ε],
and let Jδ be a smooth function of
x and t with integral 1 and support in
[- δ, δ]×[- δ, δ]; δ and ε are
positive numbers to be specified later.
We choose
z = zεδ to satisfy
The conclusion of the theorem now follows from (2.1) and the fact that
| jε*Δu(x0, t) - Δu(x0, t)|≤Cεα,
which follows from Assumption 2.
#_no#> 1
L. A. Caffarelli and A. Friedman
Regularity of the free boundary of a gas flow in an
n-dimensional porous medium
Indiana Math. J.
29
1980
361–391
#_no#> 2
K. Hollig and M. Pilant
Regularity of the free boundary for the porous medium equation
MRC Tech. Rep. 2742
#_no#> 3
J. Jerome
Approximation of Nonlinear Evolution Systems
Academic Press
New York
1983
#_no#> 4
R. J. LeVeque
Convergence of a large time step generalization of Godunov's method
for conservation laws
Comm. Pure Appl. Math.
37
1984
463–478
#_no#> 5
A large time step generalization of Godunov's method for systems
of conservation laws
#_no#> 6
B. J. Lucier
On nonlocal monotone difference methods for scalar conservation laws
Math. Comp.
47
1986
19–36
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