%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Behavior of Piece-wise Self-similar Solutoins and A Numerical % Scheme Based on it : a scheme without time marching % % a scheme without time marching and space discretization % % Yong-Jung Kim % % Aug 2000 % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 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{amstexsi} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Macro definitions for running heads and first page % %\accepted\SINUM % %\firstpageno{4} % %\lastpageno{116} % %\issuevolume{1} % %\issuenumber{2} % %\issuemonth{February} % %\placenumber{002} % place of paper in this issue % %\issueyear{1999} % \shortauthor{Y-J Kim} % \shorttitle{A scheme based on self-similairty} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Macros specific to this paper % \font\small=cmr8 at 8pt \font\smbf=cmbx8 at 8pt \font\smit=cmssi8 at 8pt \font\smallfont=cmr8 \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\,} % \def\grad{\nabla} \def\supp{\text {supp}} \def\sign{\text {sign}} \def\loc{\text {loc}} \def\Z{{\Bbb Z}} \def\R{{\Bbb R}} \def\Rn{{\Bbb R^n}} \def\N{{\Bbb N}} \def\DD{{\Cal D}} \def\xx{{\underline x}} \define\e{\varepsilon} \define\eps{\varepsilon} \define\s{\sigma} \define\g{\gamma} \define\r{\rho} \define\sgn{\text{sgn }} %---------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Macros %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\jp12{{j+1/2}} \def\jm12{{j-1/2}} \def\sig{{\sigma}} \def\del{{\partial}} \def\GRAD{{\nabla}} \def\DIV{{\nabla \cdot}} \def\H {{\bold H}} %\def\R {{\bold R}} %%already defined as \Bbb R \def\O {{\bold O}} \def\u {{\bold u}} \def\l {{\bold l}} \def\J {{\bold J}} \def\n {{\bold n}} \def\s {{\bold s}} \def\p {{\bold p}} \def\q {{\bold q}} \def\x {{\bold x}} %\def\RD{{\R^\D}} \def\tt{{\bold t}} \def\tu{\tilde u} \def\tp{\tilde p} \def\tq{\tilde q} \def\tH{\tilde H} \def\bH{\bar H} \def\btH{\bar{\tilde{H}}} \def\xP{{x^+}} \def\xM{{x^-}} \def\xPM{{x^\pm}} \def\LL{{\Cal L}} \def\BB{{\Cal B}} \def\EE{{\Cal E}} \def\HH{{\Cal H}} \def\JJ{{\Cal J}} \def\AA{{\Cal A}} \def\SS{{\Cal S}} \def\OO{{\Cal O}} %\def\dx{{d^{^\D}\!\!\!x}} \def \dx {\Delta x} \def \dy {\Delta y} \def \dt {\Delta t} \def \tfrac#1#2{{\textstyle {#1 \over #2}}} \def \frac#1#2{{#1 \over #2}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\OneFigPlace#1#2#3#4#5#6#7{{\hbox{} \vbox to #2 true in {} \special{psfile=#1 hscale=#3 vscale=#4 hoffset=#5 voffset=#6} \centerline{#7} }} \def\TwoFigPlace#1#2#3#4#5#6#7#8#9{{\hbox{} \vbox to #3 true in {} \special{psfile=#1 hscale=#4 vscale=#5 hoffset=#6 voffset=#8} \special{psfile=#2 hscale=#4 vscale=#5 hoffset=#7 voffset=#8} \noindent{#9} \medskip }} \pageno 1 \head PIECEWISE CONSTANT SOLUTIONS\endhead \midinsert\qquad\quad\special{psfile="scheme41.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 1 \quad INITIAL DATA DISCRETIZATION} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme45.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 2 \quad GODUNOV SCHEME AT t=1.2} \medskip \endinsert \vfil\eject \head PIECEWISE SELF-SIMILAR SOLUTIONS\endhead \midinsert\qquad\quad\special{psfile="scheme44.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 3 \quad APPROXIMATION WITH PIECEWISE SELF-SIMILAR FUNCTIONS} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme46.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 4 \quad SOLUTION AT t=1.2} \medskip \endinsert \vfil\eject \midinsert\qquad\quad\special{psfile="scheme46.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 4 \quad SOLUTION AT t=1.2} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme47.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 5 \quad A MAGNIFICATION NEAR THE REAL SHOCK} \medskip \endinsert \vfil\eject \head APPROXIMATION OF INITIAL DATA WITH SELF-SIMILARITY FUNCTIONS\endhead \midinsert\qquad\quad\special{psfile="scheme42.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 6 \quad SAME AREA PRINCIPLE} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme43.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 7 \quad 10 OVERLAPING BASE FUNCTIONS} \medskip \endinsert \vfil\eject \head THE CUBIC LAW\endhead \midinsert\qquad\quad\special{psfile="scheme61.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.2 in \centerline{\bf FIGURE 8 \quad DATA DISCRETIZATION} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme62.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf FIGURE 9 \quad K-SUMMATION AT t=1.15} \medskip \endinsert \vfil\eject \head SOLUTION WITH SIGN CHANGE(BURGERS) \endhead \midinsert\qquad\quad\special{psfile="scheme52.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf Figure 10 \quad SOLUTIONS AT t=1.2} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme53.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf Figure 11 \quad A MAGNIFICATION} \medskip \endinsert \vfil\eject (\head SOLUTION WITH SIGN CHANGE(BURGERS) \endhead \midinsert\qquad\quad\special{psfile="scheme54.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf Figure 12 \quad SOLUTIONS AT t=10} \medskip \endinsert \midinsert\qquad\quad\special{psfile="scheme55.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf Figure 13 \quad SOLUTIONS AT t=30} \medskip \endinsert \vfil\eject \head SOLUTION WITH SIGN CHANGE (THE CUBIC LAW)\endhead \midinsert\qquad\quad\special{psfile="scheme63.ps" hscale=70 vscale=30 hoffset=-78 voffset=-240 } \vskip 3.3 in \centerline{\bf Figure 14 \quad EVOLUTION OF A SIGN CHANGING SOLUTION} \medskip \endinsert \vfil\eject \head 2-DIMENSIONAL BURGERS\endhead \midinsert \special{psfile="scheme201.ps" hscale=90 vscale= 50 hoffset=-90 voffset=-330 } \vskip 3.4 in \centerline{\bf Figure 15\quad INITIAL DISCRETIZATION} \endinsert \midinsert \special{psfile="scheme202.ps" hscale=90 vscale= 50 hoffset= -90 voffset=-330 } \vskip 3.4 in \centerline{\bf Figure 16\quad SOUTION AT t=3} \endinsert \vfil\eject \midinsert \special{psfile="scheme12.1ps" hscale=47 vscale=19 hoffset=-78 voffset=-170 } \special{psfile="scheme12.2ps" hscale=47 vscale=19 hoffset=-78 voffset=-320 } \special{psfile="scheme12.3ps" hscale=47 vscale=19 hoffset=-78 voffset=-470 } \special{psfile="scheme12.4ps" hscale=47 vscale=19 hoffset=-78 voffset=-620 } \special{psfile="scheme12.5ps" hscale=47 vscale=19 hoffset=160 voffset=-170 } \special{psfile="scheme12.6ps" hscale=47 vscale=19 hoffset=160 voffset=-320 } \special{psfile="scheme12.7ps" hscale=47 vscale=19 hoffset=160 voffset=-470 } \special{psfile="scheme12.8ps" hscale=47 vscale=19 hoffset=160 voffset=-620 } \vskip 8.0 in \medskip \endinsert \vfil\eject \enddocument