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diff --git a/Master/texmf-dist/doc/amstex/siam/amsamp.tex b/Master/texmf-dist/doc/amstex/siam/amsamp.tex new file mode 100644 index 00000000000..44048937dd7 --- /dev/null +++ b/Master/texmf-dist/doc/amstex/siam/amsamp.tex @@ -0,0 +1,365 @@ +% This is the sample paper for the AmSTeX SIAM style file, (amstex)siam.sty +% for use with AmSTeX version 2.1 or later and amsppt.sty, version 2.1a. +% RCS information: $Revision: 1.1 $, $Date: 93/01/25 15:33:19 $. +\input amstex +\documentstyle{amstexs1} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Macro definitions for running heads and first page % +\accepted\SIMAF % +\firstpageno{10} % +\lastpageno{12} % +\issuevolume{1} % +\issuenumber{2} % +\issuemonth{February} % +\placenumber{002} % place of paper in this issue % +\issueyear{1988} % +\shortauthor{Bradley J. Lucier and Douglas N. Arnold} % +\shorttitle{A Sample Paper} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Macros specific to this paper % +\define\loner{{L^1(\Bbb R)}} % +\define\linfr{{L^\infty(\Bbb R)}} % +\define\bvr{{\roman{BV}(\Bbb R)}} % +\define\TV{{\roman {TV}}} % +\define\sdot{\,\cdot\,} % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\topmatter +\title +A SAMPLE PAPER, WITH A RATHER LONG TITLE, TO ILLUSTRATE THE +\AmSTeX\ SIAM STYLE\footnote[\boldkey*]{Unlikely to appear.} +\endtitle +\author +BRADLEY J. LUCIER\footnote[\dag]{Department of Mathematics, Purdue University, +West Lafayette, Indiana 47907. Present address, somewhere on the beach +(lucier\@math.purdue.edu). +The work of the first author was not supported by the +Wolf Foundation.}\ and DOUGLAS N. ARNOLD\footnote[\ddag]{Department +of Mathematics, Pennsylvania State University, +University Park, Pennsylvania 16802.} +\endauthor +\abstract +This sample paper illustrates many of the amstex +macros as used with the \AmSTeX\ SIAM style file amstexsiam (version 2.0a). +The \AmSTeX\ SIAM style file, which +inputs and builds upon the amsppt style (version 2.1a or later) +of Michael Spivak, gives authors easy +access to most of the typographical constructions used in SIAM journals. +It does not address the issues of the table of contents +or tables, which must be set using more primitive \TeX\ macros. +\endabstract +\keywords +porous medium, interface curves +\endkeywords +\subjclass +65N60 +\endsubjclass +\endtopmatter +\document +\subhead 1. Introduction\endsubhead +We are concerned with numerical approximations to the so-called +porous-medium equation \cite{6}, +$$ +\alignedat2 + &u_t=\phi(u)_{xx},&&\qquad x\in\Bbb R,\quad t>0,\quad\phi(u)=u^m,\quad m>1, +\\ + &u(x,0)=u_0(x),&&\qquad x\in\Bbb R. +\endalignedat +\tag 1.1 +$$ +We assume that the initial data $u_0(x)$ has bounded support, that +$0\leq u_0\leq M$, and that $\phi(u_0)_x\in\bvr$. +It is well known that a unique solution $u(x,t)$ of (1.1) exists, +and that $u$ satisfies +$$ + 0\leq u\leq M\text{ and }\TV\phi(u(\,\cdot\,,t))_x\leq\TV\phi(u_0)_x. +\tag 1.2 +$$ +If the data has slightly more regularity, then this too is satisfied +by the solution. Specifically, if $m$ is no greater than two and +$u_0$ is Lipschitz continuous, then $u(\,\cdot\,,t)$ is also Lipschitz; +if $m$ is greater than two and $(u_0^{m-1})_x\in\linfr$, then +$(u(\,\cdot\,,t)^{m-1})_x\in\linfr$ +(see [3]). (This will follow from results presented here, also.) +We also use the fact that the solution $u$ is H\"older continuous in $t$. + +\subhead 2. $\linfr$ error bounds\endsubhead +After a simple definition, we state a theorem +that expresses the error of approximations $u^h$ in +terms of the weak truncation error $E$. +\definition{Definition 2.1}\rm A {\it definition} +is the same as a theorem set in roman +type. In version 2 of the \AmSTeX\ style file for the SIAM journals, +definitions are set with their own command. +\enddefinition +\proclaim{Theorem 2.1} +Let $\{u^h\}$ be a family of approximate solutions satisfying +the following conditions for $0\leq t\leq T${\rm:} +\roster +\item For all $x\in\Bbb R$ and positive $t$, $0\leq u^h(x,t)\leq M${\rm;} +\item Both $u$ and $u^h$ are H\"older--$\alpha$ in $x$ +for some $\alpha\in(0,1\wedge 1/(m-1))${\rm;} $u^h$ is right +continuous in $t${\rm;} +and $u^h$ is H\"older continuous in $t$ on +strips $\Bbb R\times(t^n,t^{n+1})$, with the set $\{t^n\}$ having no +limit points\/{\rm;} and +\item There exists a positive function $\omega(h,\epsilon)$ such that\/{\rm:} +whenever $\{w^\epsilon\}_{0<\epsilon\leq\epsilon_0}$ is a family of functions +in $\bold X$ for which +{\roster +\item"(a)" there is a sequence of positive numbers $\epsilon$ tending +to zero, such that for these values of +$\epsilon$, $\|w^\epsilon\|_\infty\leq 1/\epsilon$, +\item"(b)" for all positive +$\epsilon$, $\|w_x^\epsilon(\sdot,t)\|_\loner\leq 1/\epsilon^2$, and +\item"(c)" for all $\epsilon>0$, +$$ +\sup\Sb +x\in\Bbb R\\0\leq t_1,t_2\leq T\endSb +\dfrac{|w^\epsilon(x,t_2)-w^\epsilon(x,t_1)|}{|t_2-t_1|^p}\leq 1/\epsilon^2, +$$ +where $p$ is some number not exceeding $1$, +\endroster}% +then\footnote{This is an obvious ploy, but we need a footnote.} + $|E (u^h,w^\epsilon,T)|\leq\omega(h,\epsilon).$ +\item +This is the fourth item in the outer roster. +\endroster +Then, there is a constant $C=C(m,M,T)$ such that +$$\multline +\|u-u^h\|_{\infty,\Bbb R\times[0,T]}\leq C\biggl[ +\sup \biggl |\int_\Bbb R(u_0(x)-u^h(x,0)) w(x,0) \,dx\biggr|\\ ++\omega(h,\epsilon)+\epsilon^\alpha\biggr],\endmultline +\tag 2.1 +$$ +where the supremum is taken over all $w\in\bold X$. +\endproclaim + +\demo{Proof} +We assume first that $Q$ is decreasing and consider the following cases: +\case{Case\/ {\rm1:} +$b'\geq 1/2$} We have $P(1/8)\geq\delta>0$ where $\delta$ +depends only on $d$, for otherwise by (3.7) applied to $P$ and $p=\infty$, +$P$ could not attain the value $1$ at $x=1$. Similarly, for +$m=(a'+b')/2$, $Q(m)\geq\delta'>0$ for some $\delta'$ depending only on $d$ +since otherwise $Q$ cannot attain the value $1$ at $x=a'$. Hence, for +$\delta''=\min(\delta,\delta')$, +$|A(y)|\geq|m-1/8|\geq b'/4\geq\frac18\max(b',1)$ for +$y\in[0,\delta'']$. On the other hand, +$|A(y)|\leq \max(b',1)$ for all $y\in[0,1]$, so (4.2) follows for +all $1\leq p\leq\infty$. +\endcase +\case{Case\/ {\rm2:} +$b'\leq 1/2$} We have $P(3/4)\leq\delta<1$ with $\delta$ +depending only on $d$ for otherwise (3.7) applied to $1-P$ and $p=\infty$ +would show that $P$ could not attain the value $0$ at $x=0$. It follows +that $|A(y)|\geq 3/4-b'\geq 1/4$, $y\in[\delta,1]$, while $|A(y)|\leq 1$ +for all $y\in[0,1]$. Hence (4.2) follows for +all $1\leq p\leq\infty$. +\endcase +We consider now when $Q$ is increasing. We can assume that $Q$ is not +a translate of $P$, i.e\., we do not have $P(x)=Q(x+\delta)$ for some $\delta$, +for then (4.2) follows trivially. In what follows, $C$ and $\delta$ +depend on $d$, and $C$ may depend on $p$. We consider the following cases: +\case{Case\/ {\rm3:} $a'\geq 1/4$ and $b'\leq 100$} +From (3.7) for $P$ +and $p=\infty$, it follows that $P(1/8)\geq\delta$ since otherwise $P$ cannot +attain the value $1$ at $x=1$. Hence $|A(y)|\geq a'-1/8\geq1/8$ on +$[0,\delta]$. On the other hand $|A(y)|\leq b'$ for all $y\in[0,1]$ and hence +(4.2) follows for all $1\leq p\leq\infty$. +\endcase +Let $z$ be in $\bold X$. Because $E(u,\sdot,\sdot)\equiv0$, +Equation (1.5) implies that +$$ +\int_\Bbb R\Delta uz|^T_0dx=\int_0^T\int_\Bbb R +\Delta u(z_t+\phi[u,u^h]z_{xx})\,dx\,dt- +E(u^h,z,t), +\tag 2.2 +$$ +where $\Delta u=u-u^h$ and +$$ +\phi[u,u^h]=\dfrac{\phi(u)-\phi(u^h)}{u-u^h}. +$$ +Extend $\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,0)$ for negative $t$, and +$\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,T)$ +for $t>T$. +Fix a point $x_0$ and a number $\epsilon>0$. Let $j_\epsilon$ +be a smooth function of $x$ with integral $1$ and support in +$[-\epsilon,\epsilon]$, +and let $J_\delta$ be a smooth function of +$x$ and $t$ with integral $1$ and support in +$[-\delta,\delta]\times[-\delta,\delta]$; $\delta$ and $\epsilon$ are +positive numbers to be specified later. +We choose $z=z^{\epsilon\delta}$ to satisfy +$$ +\aligned + &z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\Bbb R,\;0 +\leq t\leq T,\\ + &z(x,T)=j_\epsilon(x-x_0). +\endaligned +\tag 2.3 +$$ +The conclusion of the theorem now follows from (2.1) and the fact that +$$ +|j_\epsilon*\Delta u(x_0,t)-\Delta u(x_0,t)|\leq C\epsilon^\alpha, +$$ +which follows from Assumption 2. +\qquad\qed +\enddemo +\example{Example\/ {\rm 1}} This is an example of an example. +\endexample +\remark{Remark\/ {\rm 1}} Examples are set the same as definitions in +some styles, +and the same as proofs in others. What convention does this style follow? +\endremark +Sometimes you want to include a figure, as in Fig.~1. +\topinsert +\def\Bif{{\bf if\/ }}\def\Bwhile{{\bf while\/ }}\def\Belse{{\bf else\/ }} +\settabs\+\qquad&\qquad&\qquad&\qquad&\cr +\+\smc Tree Partition Algorithm \{\cr +\+&Let stack size denote the number of nodes in the\cr +\+&&subtrees stored temporarily on the local stack\cr +\+&pop I from global stack\cr +\+&set stack size := 0\cr +\+&\Bwhile (stack size $\leq$ max size and stack size + +I$\rightarrow$tree size $>$ 3 (max size)) \{\cr +\+&&process I as an interior node\cr +\+&&let min tree be the smaller of the subtrees of the two children of I\cr +\+&&let max tree be the larger of the subtrees of the two children of I\cr +\+&&\Bif (min tree$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr +\+&&&push min tree onto the global stack\cr +\+&&\} \Belse \{\cr +\+&&&push min tree onto the local stack\cr +\+&&&set stack size := stack size + min tree$\rightarrow$tree size\cr +\+&&\}\cr +\+&&set I := max tree\cr +\+&\}\cr +\+&\Bif (I$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr +\+&&push I onto the global stack\cr +\+&\} \Belse \{\cr +\+&&push I onto the local stack\cr +\+&\}\cr +\+&Process all subtrees on the local stack\cr +\+\}\cr +\botcaption{Fig.~1} Tree partition algorithm Tree partition algorithm +Tree partition algorithm Tree partition algorithm Tree partition algorithm +Tree partition algorithm Tree partition algorithm.\endcaption +\endinsert + +We finish with a table of all SIAM journals. +\midinsert +\topcaption{Table 1}{SIAM journal acronyms and titles}\endcaption +\settabs\+\indent&Acronym\indent&Title&\cr +\hbox to \hsize{\hrulefill} +\+&Acronym&Title&\cr +\hbox to \hsize{\hrulefill} +\+&SINUM&SIAM Journal on Numerical Analysis&\cr +\+&SIREV&SIAM Review&\cr +\+&SIMA&SIAM Journal on Mathematical Analysis&\cr +\+&SIMAX&SIAM Journal on Matrix Analysis and Applications&\cr +\+&SICOMP&SIAM Journal on Computing&\cr +\+&SISC&SIAM Journal on Scientific Computing&\cr +\+&SIOPT&SIAM Journal on Optimization&\cr +\+&SIAP&SIAM Journal on Applied Mathematics&\cr +\+&SICON&SIAM Journal on Control and Optimization&\cr +\+&SIDMA&SIAM Journal on Discrete Mathematics&\cr +\+&TVP&Theory of Probability and Its Applications&\cr +\hbox to \hsize{\hrulefill} +\endinsert + +\Refs +\ref + \no 1 + \by L. A. Caffarelli and A. Friedman + \paper Regularity of the free boundary of a gas flow in an + $n$-dimensional porous medium + \jour Indiana Math. J. + \vol 29 + \yr 1980 + \pages 361--391 +\endref +\ref\no 2 + \by R. DeVore and B. Lucier + \paper High order regularity for solutions of the inviscid Burgers equation + \inbook Nonlinear Hyperbolic Problems +\procinfo Proceedings of an Advanced Research Workshop, Bordeaux, +France, June 1988 + \bookinfo Lecture Notes in Mathematics + \vol 1402 + \eds C. Carasso, P. Charrier, B. Hanouzet, and J.-L. Joly + \yr 1989 + \publ Springer-Verlag + \publaddr New York + \pages 147--154 +\endref +\ref \no 3 + \bysame + \paper Wavelets + \jour Acta Numerica + \yr 1992 + \ed A. Iserles + \publ Cambridge University Press + \publaddr New York + \pages 1--56 +\endref +\ref \no 4 + \by R. A. DeVore and V. A. Popov + \paper Interpolation spaces and non-linear approximation + \inbook Function Spaces and Applications + \bookinfo Lecture Notes in Mathematics + \procinfo Proceedings of the US--Swedish Seminar held in Lund, +Sweden, June 15--21, 1986 + \vol 1302 + \eds M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin + \publ Springer-Verlag + \publaddr New York + \yr 1988 + \pages 191--205 + \endref +\ref \no 5 + \by R. A. DeVore and X. M. Yu + \paper Nonlinear $n$-widths in Besov spaces + \inbook Approximation Theory VI: Vol. 1 + \eds C. K. Chui, L. L. Schumaker, and J. D. Ward + \publ Academic Press + \publaddr New York + \yr 1989 + \pages 203--206 + \lang In Russian + \endref +\ref + \no 6 + \by K. Hollig and M. Pilant + \paper Regularity of the free boundary for the porous medium equation + \paperinfo MRC Tech. Rep. 2742 +\endref +\ref + \no 7 + \by J. Jerome + \book Approximation of Nonlinear Evolution Systems + \publ Academic Press + \publaddr New York + \yr 1983 +\endref +\ref + \no 8 + \manyby R. J. LeVeque + \paper Convergence of a large time step generalization of Godunov's method + for conservation laws + \jour Comm. Pure Appl. Math. + \vol 37 + \yr 1984 + \pages 463--478 +\endref +\ref\no 9 + \by O. Rioul and M. Vetterli + \paper Wavelets and signal processing + \jour IEEE Signal Processing Magazine + \vol 8 + \issue 4 + \yr 1991 + \toappear +\endref +\endRefs +\enddocument + |