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=1 =amstex

R L^1() L^;SPMamp;infin#infty;() BV() 0 ;SPMamp;nbsp;;SPMamp;#183;;SPMamp;nbsp; BRADLEY J. LUCIER;SPMlt;A NAME=;SPMquot;tex2html2;SPMquot; HREF=;SPMquot;footnode_mn.html#foot226;SPMquot; TARGET=;SPMquot;footer;SPMquot;;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;IMG ALT=;SPMquot;[*];SPMquot; SRC=;SPMquot;footnote.png;SPMquot;;SPMgt;;SPMlt;/SUP;SPMgt;;SPMlt;/A;SPMgt;;SPMquot;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.57ex; ;SPMquot; SRC=;SPMquot;img1.png;SPMquot; ALT=;SPMquot;#tex2html_nomath_inline194#;SPMquot;;SPMgt;;SPMquot;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;SPMlt;A NAME=;SPMquot;tex2html3;SPMquot; HREF=;SPMquot;footnode_mn.html#foot227;SPMquot; TARGET=;SPMquot;footer;SPMquot;;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;IMG ALT=;SPMquot;[*];SPMquot; SRC=;SPMquot;footnote.png;SPMquot;;SPMgt;;SPMlt;/SUP;SPMgt;;SPMlt;/A;SPMgt;;SPMquot;;SPMlt;IMG STYLE=;SPMquot;height: 2.24ex; vertical-align: 176.61ex; ;SPMquot; SRC=;SPMquot;img2.png;SPMquot; ALT=;SPMquot;#tex2html_nomath_inline196#;SPMquot;;SPMgt;;SPMquot;Department of Mathematics, University of Maryland, College Park, Maryland 20742.

;SPMlt;H3;SPMgt;Abstract:;SPMlt;/H3;SPMgt; ;SPMlt;DIV;SPMgt; This nonsense paper exemplifies the IMA preprint style file for ;SPMlt;i;SPMgt;AmS-TeX;SPMlt;/i;SPMgt;;SPMamp;nbsp; imappt.sty. The IMA preprint style file, which is used by the Institute for Mathematics and its Applications for its preprints, is offered as a general purpose preprint style for mathematical papers. It is a modification of the amsppt style of Michael Spivak. This paper also serves to illustrate many of the amstex macros as used with the imappt style file.;SPMlt;/DIV;SPMgt; ;SPMlt;P;SPMgt;

;SPMlt;P;SPMgt; 1. Introduction We are concerned with numerical approximations to the so-called porous-medium equation [#!5!#], ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math1#

#tex2html_wrap_indisplay233#2 ;SPMamp; ut = φ(u)xx, ;SPMamp; ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;x#tex2html_wrap_indisplay234#,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;t ;SPMgt; 0,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;φ(u) = um,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;m ;SPMgt; 1,
;SPMamp; u(x, 0) = u0(x), ;SPMamp; ;SPMamp; ;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;x#tex2html_wrap_indisplay235#.#tex2html_wrap_indisplay236##tex2html_wrap_indisplay237#1.1

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; ;SPMlt;IMG STYLE=;SPMquot;height: 2.17ex; vertical-align: 176.55ex; ;SPMquot; SRC=;SPMquot;img3.png;SPMquot; ALT=;SPMquot;#math2##tex2html_wrap_indisplay239#2 ;SPMamp; amp;ut = φ(u)xx, ;SPMamp; amp; ;SPMamp; amp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;x#tex2html_wrap_indisplay240#,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;t ;SPMamp; gt;0,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;...... = um,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;m ;SPMamp; gt;1,
;SPMamp; amp;u(x, 0) = u0(x), ;SPMamp; amp; ;SPMamp; amp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;x#tex2html_wrap_indisplay241#.#tex2html_wrap_indisplay242##tex2html_wrap_indisplay243#1.1;SPMquot;;SPMgt; ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; We assume that the initial data ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;) has bounded support, that ;SPMlt;!-- MATH #math3#0≤u0M --;SPMgt; 0;SPMamp;#8804;;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;;SPMamp;#8804;;SPMlt;I;SPMgt;M;SPMlt;/I;SPMgt;, and that ;SPMlt;!-- MATH #math4#φ(u0)x#tex2html_wrap_inline246# --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;);SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;;SPMamp;#8712;;SPMlt;IMG STYLE=;SPMquot;height: 2.38ex; vertical-align: 176.85ex; ;SPMquot; SRC=;SPMquot;img4.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline248#;SPMquot;;SPMgt;. It is well known that a unique solution ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;) of (1.1) exists, and that ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt; satisfies ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math5#

0≤uM and #tex2html_wrap_indisplay250#φ(u( ⋅ , t))x#tex2html_wrap_indisplay251#φ(u0)x.#tex2html_wrap_indisplay252#1.2

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; 0;SPMamp;#8804;;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMamp;#8804;;SPMlt;I;SPMgt;M;SPMlt;/I;SPMgt; and ;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img5.png;SPMquot; ALT=;SPMquot;#math6##tex2html_wrap_indisplay254#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;(;SPMamp;nbsp;;SPMamp;#8901;;SPMamp;nbsp;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;));SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;;SPMamp;#8804;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img5.png;SPMquot; ALT=;SPMquot;#math7##tex2html_wrap_indisplay256#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;);SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;.;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img6.png;SPMquot; ALT=;SPMquot;#math8##tex2html_wrap_indisplay258#;SPMquot;;SPMgt;1.2 ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; If the data has slightly more regularity, then this too is satisfied by the solution. Specifically, if ;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; is no greater than two and ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt; is Lipschitz continuous, then ;SPMlt;!-- MATH #math9#u( ⋅ , t) --;SPMgt; ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;(;SPMamp;nbsp;;SPMamp;#8901;;SPMamp;nbsp;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;) is also Lipschitz; if ;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; is greater than two and ;SPMlt;!-- MATH #math10#(u0m-1)x#tex2html_wrap_inline261# --;SPMgt; (;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;;SPMlt;SUP;SPMgt;m-1;SPMlt;/SUP;SPMgt;);SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;;SPMamp;#8712;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img7.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline263#;SPMquot;;SPMgt;, then ;SPMlt;!-- MATH #math11#(u( ⋅ , t)m-1)x#tex2html_wrap_inline265# --;SPMgt; (;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;(;SPMamp;nbsp;;SPMamp;#8901;;SPMamp;nbsp;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;);SPMlt;SUP;SPMgt;m-1;SPMlt;/SUP;SPMgt;);SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;;SPMamp;#8712;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img7.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline267#;SPMquot;;SPMgt; (see [3]). (This will follow from results presented here, also.) We also use the fact that the solution ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt; is H;SPMamp;#246;lder continuous in ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;.

;SPMlt;P;SPMgt; 2. ;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img7.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline269#;SPMquot;;SPMgt; error bounds After a simple definition, we state a theorem that expresses the error of approximations ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt; in terms of the weak truncation error ;SPMlt;I;SPMgt;E;SPMlt;/I;SPMgt;. Definition 2.1A ;SPMlt;I;SPMgt;definition;SPMlt;/I;SPMgt; is the same as a theorem set in roman type.

Theorem 2.1 Let <#10#>;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;<#10#> be a family of approximate solutions satisfying the following conditions for ;SPMlt;!-- MATH #math12#0≤tT --;SPMgt; 0;SPMamp;#8804;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMamp;#8804;;SPMlt;I;SPMgt;T;SPMlt;/I;SPMgt;:

For all ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMamp;#8712;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img8.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline272#;SPMquot;;SPMgt; and positive ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;, ;SPMlt;!-- MATH #math13#0≤uh(x, t)≤M --;SPMgt; 0;SPMamp;#8804;;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;);SPMamp;#8804;;SPMlt;I;SPMgt;M;SPMlt;/I;SPMgt;; Both ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt; and ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt; are H;SPMamp;#246;lder;SPMamp;ndash;;SPMlt;I;SPMgt;;SPMamp;#945;;SPMlt;/I;SPMgt; in ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt; for some ;SPMlt;!-- MATH #math14#α∈(0, 1∧1/(m - 1)) --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#945;;SPMlt;/I;SPMgt;;SPMamp;#8712;(0, 1;SPMamp;#8743;1/(;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; - 1)); ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt; is right continuous in ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;; and ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt; is H;SPMamp;#246;lder continuous in ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; on strips ;SPMlt;!-- MATH #math15##tex2html_wrap_inline276#×(tn, tn+1) --;SPMgt; ;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img8.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline278#;SPMquot;;SPMgt;;SPMamp;#215;(;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;n;SPMlt;/SUP;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;n+1;SPMlt;/SUP;SPMgt;), with the set <#12#>;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;n;SPMlt;/SUP;SPMgt;<#12#> having no limit points; and There exists a positive function ;SPMlt;!-- MATH #math16#ω(h, ε) --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#969;;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;h;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;) such that: whenever ;SPMlt;!-- MATH #math17#{wε}0 ;SPMlt; εε0 --;SPMgt; <#14#>;SPMlt;I;SPMgt;w;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt;<#14#>;SPMlt;SUB;SPMgt;0 ;SPMamp;lt; ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMamp;#8804;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;;SPMlt;/SUB;SPMgt; is a family of functions in ;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img9.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline282#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;X;SPMlt;/I;SPMgt; for which

;SPMquot;(a);SPMquot; there is a sequence of positive numbers ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt; tending to zero, such that for these values of ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;, ;SPMlt;!-- MATH #math18#| wε|≤1/ε --;SPMgt; | ;SPMlt;I;SPMgt;w;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt;|;SPMlt;SUB;SPMgt;;SPMamp;#8734;;SPMlt;/SUB;SPMgt;;SPMamp;#8804;1/;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;, ;SPMquot;(b);SPMquot; for all positive ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;, ;SPMlt;!-- MATH #math19#| wxε(#tex2html_wrap_inline285#, t)|#tex2html_wrap_inline286#≤1/ε2 --;SPMgt; | ;SPMlt;I;SPMgt;w;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt;(;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img10.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline288#;SPMquot;;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;)|;SPMlt;SUB;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img11.png;SPMquot; ALT=;SPMquot;#math20##tex2html_wrap_inline290#;SPMquot;;SPMgt;;SPMlt;/SUB;SPMgt;;SPMamp;#8804;1/;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;, and ;SPMquot;(c);SPMquot; for all ;SPMlt;!-- MATH #math21#ε ;SPMgt; 0 --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt; ;SPMamp;gt; 0, ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math22#

sup#tex2html_wrap_indisplay293#x#tex2html_wrap_indisplay294#
0≤t1, t2T#tex2html_wrap_indisplay295##tex2html_wrap_indisplay296#≤1/ε2,

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; sup;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img12.png;SPMquot; ALT=;SPMquot;#math23##tex2html_wrap_indisplay298#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMamp;#8712;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img13.png;SPMquot; ALT=;SPMquot;#math24##tex2html_wrap_indisplay300#;SPMquot;;SPMgt; ;SPMlt;BR;SPMgt; 0;SPMamp;#8804;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;1;SPMlt;/SUB;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;2;SPMlt;/SUB;SPMgt;;SPMamp;#8804;;SPMlt;I;SPMgt;T;SPMlt;/I;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img12.png;SPMquot; ALT=;SPMquot;#math25##tex2html_wrap_indisplay302#;SPMquot;;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.70ex; ;SPMquot; SRC=;SPMquot;img14.png;SPMquot; ALT=;SPMquot;#math26##tex2html_wrap_indisplay304#;SPMquot;;SPMgt;;SPMamp;#8804;1/;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;, ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; where ;SPMlt;I;SPMgt;p;SPMlt;/I;SPMgt; is some number not exceeding 1,

then ;SPMlt;!-- MATH #math27#| E(uh, wε, T)|≤ω(h, ε). --;SPMgt; | ;SPMlt;I;SPMgt;E;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;, ;SPMlt;I;SPMgt;w;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt;, ;SPMlt;I;SPMgt;T;SPMlt;/I;SPMgt;)|;SPMamp;#8804;;SPMlt;I;SPMgt;;SPMamp;#969;;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;h;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;).

Then, there is a constant ;SPMlt;!-- MATH #math28#C = C(m, M, T) --;SPMgt; ;SPMlt;I;SPMgt;C;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;C;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;M;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;T;SPMlt;/I;SPMgt;) such that ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math29#

| u - uh|∞,#tex2html_wrap_indisplay308#×[0, T]C#tex2html_wrap_indisplay309#sup#tex2html_wrap_indisplay310##tex2html_wrap_indisplay311#(u0(x) - uh(x, 0))w(x, 0) dx#tex2html_wrap_indisplay312# + ω(h, ε) + εα#tex2html_wrap_indisplay313#,#tex2html_wrap_indisplay314#2.1

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; | ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt; - ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;|;SPMlt;SUB;SPMgt;;SPMamp;#8734;,;SPMlt;IMG STYLE=;SPMquot;height: 2.38ex; vertical-align: 176.85ex; ;SPMquot; SRC=;SPMquot;img15.png;SPMquot; ALT=;SPMquot;#math30##tex2html_wrap_inline316#;SPMquot;;SPMgt;;SPMamp;#215;[0, T];SPMlt;/SUB;SPMgt;;SPMamp;#8804;;SPMlt;I;SPMgt;C;SPMlt;/I;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -3.02ex; ;SPMquot; SRC=;SPMquot;img16.png;SPMquot; ALT=;SPMquot;#math31##tex2html_wrap_indisplay318##tex2html_wrap_indisplay319#sup#tex2html_wrap_indisplay320##tex2html_wrap_indisplay321#(u0(x) - uh(x, 0))w(x, 0) dx#tex2html_wrap_indisplay322#+ω(h, ε)+εα;SPMquot;;SPMgt;sup;SPMlt;IMG STYLE=;SPMquot;height: 7.06ex; vertical-align: 169.33ex; ;SPMquot; SRC=;SPMquot;img17.png;SPMquot; ALT=;SPMquot;#math32##tex2html_wrap_indisplay324##tex2html_wrap_indisplay325##tex2html_wrap_indisplay326#(u0(x)-uh(x, 0))w(x, 0) dx;SPMquot;;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 7.06ex; vertical-align: 170.12ex; ;SPMquot; SRC=;SPMquot;img18.png;SPMquot; ALT=;SPMquot;#math33##tex2html_wrap_indisplay328#;SPMquot;;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;) - ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;, 0));SPMlt;I;SPMgt;w;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;, 0);SPMamp;nbsp;d;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 5.18ex; vertical-align: 171.15ex; ;SPMquot; SRC=;SPMquot;img19.png;SPMquot; ALT=;SPMquot;#math34##tex2html_wrap_indisplay330##tex2html_wrap_indisplay331##tex2html_wrap_indisplay332#(u0(x)-uh(x, 0))w(x, 0) dx#tex2html_wrap_indisplay333#;SPMquot;;SPMgt; + ;SPMlt;I;SPMgt;;SPMamp;#969;;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;h;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;) + ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#945;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 7.06ex; vertical-align: 169.33ex; ;SPMquot; SRC=;SPMquot;img20.png;SPMquot; ALT=;SPMquot;#math35##tex2html_wrap_indisplay335##tex2html_wrap_indisplay336#sup#tex2html_wrap_indisplay337##tex2html_wrap_indisplay338#(u0(x) - uh(x, 0))w(x, 0) dx#tex2html_wrap_indisplay339#+ω(h, ε)+εα#tex2html_wrap_indisplay340#;SPMquot;;SPMgt;,;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img6.png;SPMquot; ALT=;SPMquot;#math36##tex2html_wrap_indisplay342#;SPMquot;;SPMgt;2.1 ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; where the supremum is taken over all ;SPMlt;!-- MATH #math37#w#tex2html_wrap_inline344#X --;SPMgt; ;SPMlt;I;SPMgt;w;SPMlt;/I;SPMgt;;SPMamp;#8712;;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img9.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline346#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;X;SPMlt;/I;SPMgt;.

ProofLet ;SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt; be in ;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img9.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline348#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;X;SPMlt;/I;SPMgt;. Because ;SPMlt;!-- MATH #math38#E(u,#tex2html_wrap_inline350#,#tex2html_wrap_inline351#)≡ 0 --;SPMgt; ;SPMlt;I;SPMgt;E;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;,;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img10.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline353#;SPMquot;;SPMgt;,;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img10.png;SPMquot; ALT=;SPMquot;#tex2html_wrap_inline355#;SPMquot;;SPMgt;);SPMamp;#8801; 0, Equation (1.5) implies that ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math39#

#tex2html_wrap_indisplay357#Δuz|T0dx = #tex2html_wrap_indisplay358##tex2html_wrap_indisplay359#Δu(zt + φ[u, uh]zxx) dx dt - E(uh, z, t),#tex2html_wrap_indisplay360#2.2

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; ;SPMlt;IMG STYLE=;SPMquot;height: 7.06ex; vertical-align: 170.12ex; ;SPMquot; SRC=;SPMquot;img18.png;SPMquot; ALT=;SPMquot;#math40##tex2html_wrap_indisplay362#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;uz;SPMlt;/I;SPMgt;|;SPMlt;SUP;SPMgt;T;SPMlt;/SUP;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;dx;SPMlt;/I;SPMgt; = ;SPMlt;IMG STYLE=;SPMquot;height: 7.06ex; vertical-align: 170.12ex; ;SPMquot; SRC=;SPMquot;img21.png;SPMquot; ALT=;SPMquot;#math41##tex2html_wrap_indisplay364#;SPMquot;;SPMgt;;SPMlt;IMG STYLE=;SPMquot;height: 7.06ex; vertical-align: 170.12ex; ;SPMquot; SRC=;SPMquot;img18.png;SPMquot; ALT=;SPMquot;#math42##tex2html_wrap_indisplay366#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;u;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;t;SPMlt;/SUB;SPMgt; + ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;[;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;];SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;xx;SPMlt;/SUB;SPMgt;);SPMamp;nbsp;d;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMamp;nbsp;d;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; - ;SPMlt;I;SPMgt;E;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;, ;SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;),;SPMlt;IMG STYLE=;SPMquot;height: 196.25ex; vertical-align: -0.12ex; ;SPMquot; SRC=;SPMquot;img6.png;SPMquot; ALT=;SPMquot;#math43##tex2html_wrap_indisplay368#;SPMquot;;SPMgt;2.2 ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; where ;SPMlt;!-- MATH #math44#Δu = u - uh --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#916;u;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt; - ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt; and ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math45#

φ[u, uh] = #tex2html_wrap_indisplay371#.

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;[;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;] = ;SPMlt;IMG STYLE=;SPMquot;height: 5.87ex; vertical-align: 172.78ex; ;SPMquot; SRC=;SPMquot;img22.png;SPMquot; ALT=;SPMquot;#math46##tex2html_wrap_indisplay373#;SPMquot;;SPMgt;. ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; Extend ;SPMlt;!-- MATH #math47#φ[u, uh](⋅, t) = φ[u, uh](⋅, 0) --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;[;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;](;SPMamp;#8901;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;) = ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;[;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;](;SPMamp;#8901;, 0) for negative ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;, and ;SPMlt;!-- MATH #math48#φ[u, uh](⋅, t) = φ[u, uh](⋅, T) --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;[;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;](;SPMamp;#8901;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;) = ;SPMlt;I;SPMgt;;SPMamp;#966;;SPMlt;/I;SPMgt;[;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;u;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;h;SPMlt;/SUP;SPMgt;](;SPMamp;#8901;, ;SPMlt;I;SPMgt;T;SPMlt;/I;SPMgt;) for ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; ;SPMamp;gt; ;SPMlt;I;SPMgt;T;SPMlt;/I;SPMgt;.;SPMlt;A NAME=;SPMquot;tex2html4;SPMquot; HREF=;SPMquot;footnode_mn.html#foot33;SPMquot; TARGET=;SPMquot;footer;SPMquot;;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;IMG ALT=;SPMquot;[*];SPMquot; SRC=;SPMquot;footnote.png;SPMquot;;SPMgt;;SPMlt;/SUP;SPMgt;;SPMlt;/A;SPMgt;Fix a point ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt; and a number ;SPMlt;!-- MATH #math49#ε ;SPMgt; 0 --;SPMgt; ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt; ;SPMamp;gt; 0. Let ;SPMlt;!-- MATH #math50#jε --;SPMgt; ;SPMlt;I;SPMgt;j;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;/SUB;SPMgt; be a smooth function of ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt; with integral 1 and support in ;SPMlt;!-- MATH #math51#[- ε, ε] --;SPMgt; [- ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;], and let ;SPMlt;I;SPMgt;J;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#948;;SPMlt;/I;SPMgt;;SPMlt;/SUB;SPMgt; be a smooth function of ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt; and ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; with integral 1 and support in ;SPMlt;!-- MATH #math52#[- δ, δ]×[- δ, δ] --;SPMgt; [- ;SPMlt;I;SPMgt;;SPMamp;#948;;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;;SPMamp;#948;;SPMlt;/I;SPMgt;];SPMamp;#215;[- ;SPMlt;I;SPMgt;;SPMamp;#948;;SPMlt;/I;SPMgt;, ;SPMlt;I;SPMgt;;SPMamp;#948;;SPMlt;/I;SPMgt;]; ;SPMlt;I;SPMgt;;SPMamp;#948;;SPMlt;/I;SPMgt; and ;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt; are positive numbers to be specified later. We choose ;SPMlt;!-- MATH #math53#z = zεδ --;SPMgt; ;SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMamp;#948;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt; to satisfy ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math54#

#tex2html_wrap_indisplay382# ;SPMamp; zt + (δ + Jδ*φ[u, uh])zxx = 0,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;x#tex2html_wrap_indisplay383#,  0≤tT,
;SPMamp; z(x, T) = jε(x - x0).#tex2html_wrap_indisplay384##tex2html_wrap_indisplay385#2.3

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; ;SPMlt;IMG STYLE=;SPMquot;height: 2.80ex; vertical-align: 175.92ex; ;SPMquot; SRC=;SPMquot;img23.png;SPMquot; ALT=;SPMquot;#math55##tex2html_wrap_indisplay387# ;SPMamp; amp;zt + (δ + Jδ*φ[u, uh])zxx = 0,;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;x#tex2html_wrap_indisplay388#,  0≤tT,
;SPMamp; amp;z(x, T) = jε(x - x0).#tex2html_wrap_indisplay389##tex2html_wrap_indisplay390#2.3;SPMquot;;SPMgt; ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; The conclusion of the theorem now follows from (2.1) and the fact that ;SPMlt;P;SPMgt;;SPMlt;!-- MATH

#math56#

| jε*Δu(x0, t) - Δu(x0, t)|≤α,

--;SPMgt; ;SPMlt;/P;SPMgt; ;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt; | ;SPMlt;I;SPMgt;j;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;/SUB;SPMgt;*;SPMlt;I;SPMgt;;SPMamp;#916;u;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;) - ;SPMlt;I;SPMgt;;SPMamp;#916;u;SPMlt;/I;SPMgt;(;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;, ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;)|;SPMamp;#8804;;SPMlt;I;SPMgt;C;SPMamp;#949;;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#945;;SPMlt;/I;SPMgt;;SPMlt;/SUP;SPMgt;, ;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt; which follows from Assumption 2.;SPMlt;IMG STYLE=;SPMquot;height: 2.80ex; vertical-align: 176.50ex; ;SPMquot; SRC=;SPMquot;img24.png;SPMquot; ALT=;SPMquot;#math57##tex2html_wrap_inline393#;SPMquot;;SPMgt;

;SPMlt;A HREF=;SPMlt;tex2html_cr_mark;SPMgt;#_no#;SPMgt;;SPMlt;tex2html_cr_mark;SPMgt;;SPMlt;/A;SPMgt; 1 K. Hollig and M. Pilant Regularity of the free boundary for the porous medium equation MRC Tech. Rep. 2742

;SPMlt;A HREF=;SPMlt;tex2html_cr_mark;SPMgt;#_no#;SPMgt;;SPMlt;tex2html_cr_mark;SPMgt;;SPMlt;/A;SPMgt; 2 J. Jerome Approximation of Nonlinear Evolution Systems Academic Press New York 1983

;SPMlt;A HREF=;SPMlt;tex2html_cr_mark;SPMgt;#_no#;SPMgt;;SPMlt;tex2html_cr_mark;SPMgt;;SPMlt;/A;SPMgt; 3 R. J. LeVeque Convergence of a large time step generalization of Godunov's method for conservation laws Comm. Pure Appl. Math. 37 1984 463;SPMamp;ndash;478

;SPMlt;A HREF=;SPMlt;tex2html_cr_mark;SPMgt;#_no#;SPMgt;;SPMlt;tex2html_cr_mark;SPMgt;;SPMlt;/A;SPMgt; 4

A large time step generalization of Godunov's method for systems of conservation laws

;SPMlt;A HREF=;SPMlt;tex2html_cr_mark;SPMgt;#_no#;SPMgt;;SPMlt;tex2html_cr_mark;SPMgt;;SPMlt;/A;SPMgt; 5 B. J. Lucier On nonlocal monotone difference methods for scalar conservation laws Math. Comp. 47 1986 19;SPMamp;ndash;36

;''ARRAY(0x1e0a0158) ;SPMlt;HR;SPMgt;

;SPMlt;/BODY;SPMgt; ;SPMlt;/HTML;SPMgt;