1、对流方程偏微分方程的数值解法用迎风格式解对流方程function u = peYF(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);if a0for j=1:(n+M)u0(j) = IniU(minx+(j-M-1)*h);endelse for j=1:(n+M)u0(j) = IniU(minx+(j-1)*h);endend u1 = u0;for k=1:Mif a0 for i=(k+1):n+Mu1(i) = -dt*a*(u0(i)-u0(i-1)/h+u0(i);endelsefor i=1:n+M-ku1(i) =
2、-dt*a*(u0(i+1)-u0(i)/h+u0(i);endendu0 = u1;endif a0u = u1(M+1):M+n);elseu = u1(1:n);endformat long;用拉克斯-弗里德里希斯格式解对流方程function u = peHypbLax(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);for j=1:(n+2*M)u0(j) = IniU(minx+(j-M-1)*h);end u1 = u0;for k=1:Mfor i=k+1:n+2*M-ku1(i) = -dt*a*(u0(i+1)-u0
3、(i-1)/h/2+(u0(i+1)+u0(i-1)/2;endu0 = u1;endu = u1(M+1):(M+n);format short;用拉克斯-温德洛夫格式解对流方程function u = peLaxW(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);for j=1:(n+2*M)u0(j) = IniU(minx+(j-M-1)*h);end u1 = u0;for k=1:Mfor i=k+1:n+2*M-ku1(i) = dt*dt*a*a*(u0(i+1)-2*u0(i)+u0(i-1)/2/h/h - .dt*
4、a*(u0(i+1)-u0(i-1)/h/2+u0(i);endu0 = u1;endu = u1(M+1):(M+n);format short;用比姆-沃明格式解对流方程function u = peBW(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);for j=1:(n+2*M)u0(j) = IniU(minx+(j-2*M-1)*h);end u1 = u0;for k=1:Mfor i=2*k+1:n+2*Mu1(i) = u0(i)-dt*a*(u0(i)-u0(i-1)/h-a*dt*(1-a*dt/h)* .(u0(
5、i)-2*u0(i-1)+u0(i-2)/2/h;endu0 = u1;endu = u1(2*M+1):(2*M+n);format short;用 Richtmyer 多步格式解对流方程function u = peRich(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);for j=1:(n+4*M)u0(j) = IniU(minx+(j-2*M-1)*h);end u1 = u0;for k=1:Mfor i=2*k+1:n+4*M-2*ktmpU1 = -dt*a*(u0(i+2)-u0(i)/h/4+(u0(i+2)+u0
6、(i)/2;tmpU2 = -dt*a*(u0(i)-u0(i-2)/h/4+(u0(i)+u0(i-2)/2; u1(i) = -dt*a*(tmpU1-tmpU2)/h/2+u0(i);endu0 = u1;endu = u1(2*M+1):(2*M+n);format short;用拉克斯-温德洛夫多步格式解对流方程function u = peMLW(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);for j=1:(n+2*M)u0(j) = IniU(minx+(j-M-1)*h);end u1 = u0;for k=1:Mf
7、or i=k+1:n+2*M-ktmpU1 = -dt*a*(u0(i+1)-u0(i)/h/2+(u0(i+1)+u0(i)/2;tmpU2 = -dt*a*(u0(i)-u0(i-1)/h/2+(u0(i)+u0(i-1)/2; u1(i) = -dt*a*(tmpU1-tmpU2)/h+u0(i);endu0 = u1;endu = u1(M+1):(M+n);format short;用 MacCormack 多步格式解对流方程function u = peMC(a,dt,n,minx,maxx,M)format long;h = (maxx-minx)/(n-1);for j=1:(
8、n+2*M)u0(j) = IniU(minx+(j-M-1)*h);end u1 = u0;for k=1:Mfor i=k+1:n+2*M-ktmpU1 = -dt*a*(u0(i+1)-u0(i)/h+u0(i);tmpU2 = -dt*a*(u0(i)-u0(i-1)/h+u0(i-1); u1(i) = -dt*a*(tmpU1-tmpU2)/h/2+(u0(i)+tmpU1)/2;endu0 = u1;endu = u1(M+1):(M+n);format short;用拉克斯-弗里德里希斯格式解二维对流方程的初值问题function u = pe2LF(a,b,dt,nx,min
9、x,maxx,ny,miny,maxy,M)%啦- 佛format long;hx = (maxx-minx)/(nx-1);hy = (maxy-miny)/(ny-1);for i=1:nx+2*Mfor j=1:(ny+2*M)u0(i,j) = Ini2U(minx+(i-M-1)*hx,miny+(j-M-1)*hy);endend u1 = u0;for k=1:Mfor i=k+1:nx+2*M-kfor j=k+1:ny+2*M-ku1(i,j) = (u0(i+1,j)+u0(i-1,j)+u0(i,j+1)+u0(i,j-1)/4 .-a*dt*(u0(i+1,j)-u0(
10、i-1,j)/2/hx .-b*dt*(u0(i,j+1)-u0(i,j-1)/2/hy;endendu0 = u1;endu = u1(M+1):(M+nx),(M+1):(M+ny);format short; 用拉克斯-弗里德里希斯格式解二维对流方程的初值问题function u = pe2FL(a,b,dt,nx,minx,maxx,ny,miny,maxy,M)%近似分裂format long;hx = (maxx-minx)/(nx-1);hy = (maxy-miny)/(ny-1);for i=1:nx+4*Mfor j=1:(ny+4*M)u0(i,j) = Ini2U(mi
11、nx+(i-2*M-1)*hx,miny+(j-2*M-1)*hy);endend u1 = u0;for k=1:Mfor i=2*k+1:nx+4*M-2*kfor j=2*k-1:ny+4*M-2*k+2tmpU(i,j) = u0(i,j) - a*dt*(u0(i+1,j)-u0(i-1,j)/2/hx + .(a*dt/hx)2*(u0(i+1,j)-2*u0(i,j)+u0(i-1,j)/2;endendfor i=2*k+1:nx+4*M-2*kfor j=2*k+1:nx+4*M-2*ku1(i,j) = tmpU(i,j) - b*dt*(tmpU(i,j+1)-tmpU(i,j-1)/2/hy + .(b*dt/hy)2*(tmpU(i,j+1)-2*tmpU(i,j)+tmpU(i,j-1)/2;endendu0 = u1;endu = u1(2*M+1):(2*M+nx),(2*M+1):(2*M+ny);format short;
