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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);end else for j=1:(n+M)u0(j) = IniU(minx+(j-1)*h);end end 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
2、(i) = -dt*a*(u0(i+1)-u0(i)/h+u0(i);endendu0 = u1; end if a0u = u1(M+1):M+n); elseu = u1(1:n); end format 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) =
3、-dt*a*(u0(i+1)-u0(i-1)/h/2+(u0(i+1)+u0(i-1)/2;endu0 = u1; end u = 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*
4、u0(i)+u0(i-1)/2/h/h - .dt*a*(u0(i+1)-u0(i-1)/h/2+u0(i);endu0 = u1; end u = 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(
5、i)-u0(i-1)/h-a*dt*(1-a*dt/h)* .(u0(i)-2*u0(i-1)+u0(i-2)/2/h;endu0 = u1; end u = 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*kt
6、mpU1 = -dt*a*(u0(i+2)-u0(i)/h/4+(u0(i+2)+u0(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; end u = 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)u
7、0(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/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; end u = u1(M+1):(M+n); format short; 用 MacCormack 多步格式解对流方程function u = peMC(a,dt,n
8、,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-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; end u = u1(M+1):(M+n); format
9、 short; 用拉克斯用拉克斯-弗里德里希斯格式解二维对流方程的初值问题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,minx,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);end end u1 = u0; for k=1:Mfor i=k+1:nx+2*M
10、-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(i-1,j)/2/hx .-b*dt*(u0(i,j+1)-u0(i,j-1)/2/hy;endendu0 = u1; end u = u1(M+1):(M+nx),(M+1):(M+ny); format short; 用拉克斯用拉克斯-弗里德里希斯格式解二维对流方程的初值问题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
11、,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(minx+(i-2*M-1)*hx,miny+(j-2*M-1)*hy);end end 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; end u = u1(2*M+1):(2*M+nx),(2*M+1):(2*M+ny); format short;
