=amstex =10 R L^1() L^&infin#infty;() 0() 0  ·  BRADLEY J. LUCIER[*]"\dag"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[*]"\ddag"Department of Mathematics, University of Maryland, College Park, Maryland 20742.

Abstract:

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1. Introduction We are concerned with numerical approximations to the so-called porous-medium equation [#!6!#],

$\displaystyle \alignedat2
&u_t=\phi(u)_{xx},&&\qquad x\in\BbbR,\quad t>0,\quad...
...=u^m,\quad m>1,
\\
&u(x,0)=u_0(x),&&\qquad x\in\BbbR.
\endalignedat
\tag 1.1
$

We assume that the initial data u0(x) has bounded support, that 0≤u0M, and that φ(u0)x$\bvr$. It is well known that a unique solution u(x, t) of (1.1) exists, and that u satisfies

0≤uM and $\displaystyle \TV$φ(u( ⋅ , t))x$\displaystyle \TV$φ(u0)x.$\displaystyle \tag$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$\linfr$, then (u( ⋅ , t)m-1)x$\linfr$ (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. $\linfr$ 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≤tT: For all x$\BbbR$ 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 $\BbbR$×(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 < εε0 is a family of functions in $\bold$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($\sdot$, t)|$\scriptstyle \loner$≤1/ε2, and "(c)" for all ε > 0,

sup$\displaystyle \Sb$x$\displaystyle \BbbR$
0≤t1, t2T$\displaystyle \Sb$$\displaystyle {\dfrac{{\vert w^\epsilon(x,t_2)-w^\epsilon(x,t_1)\vert}}{{\vert t_2-t_1\vert^p}}}$≤1/ε2,

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|∞,$\scriptstyle \BbbR$×[0, T]C$\displaystyle \left[\vphantom{
\sup \left \vert\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx\right\vert+
\omega(h,\epsilon)+\epsilon^\alpha}\right.$sup$\displaystyle \left\vert\vphantom{\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx}\right.$$\displaystyle \int_{\BbbR}^{}$(u0(x) - uh(x, 0))w(x, 0) dx$\displaystyle \left.\vphantom{\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx}\right\vert$ + ω(h, ε) + εα$\displaystyle \left.\vphantom{
\sup \left \vert\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx\right\vert+
\omega(h,\epsilon)+\epsilon^\alpha}\right]$,$\displaystyle \tag$2.1

where the supremum is taken over all w$\bold$X. ProofLet z be in $\bold$X. Because E(u,$\sdot$,$\sdot$)≡ 0, Equation (1.5) implies that

$\displaystyle \int_{\BbbR}^{}$Δuz|T0dx = $\displaystyle \int_{0}^{T}$$\displaystyle \int_{\BbbR}^{}$Δu(zt + φ[u, uh]zxx) dx dt - E(uh, z, t),$\displaystyle \tag$2.2

where Δu = u - uh and

φ[u, uh] = $\displaystyle {\dfrac{{\phi(u)-\phi(u^h)}}{{u-u^h}}}$.

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

$\displaystyle \aligned
&z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\BbbR,\;0\leq t\leq T,
\\
&z(x,T)=j_\epsilon(x-x_0).
\endaligned
\tag 2.3
$

The conclusion of the theorem now follows from (2.1) and the fact that

| jε*Δu(x0, t) - Δu(x0, t)|≤α,

which follows from Assumption 2.$\qedsymbol$ #_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 ;''ARRAY(0x2ba7150)