Matlab in Physics Nguyen Van Hong code

% A few Matlab examples including 2D and 3D graphics, animations, and % matrix operations set(0,'DefaultTextInterpreter','LaTeX') %% 2d plot clear, clf x=-1:.1:1; y=x.^3; plot(x,y) box off xlabel('$x$') ylabel('$y$') title('$y=x^3$') %% 3d plot clear, clf [x y]=meshgrid(linspace(-1,1,50)); z=x.^2+y.^2; mesh(x,y,z) %% 3d plot (improvement 1) clear, clf r=linspace(0,1,25); vth=linspace(0,2*pi,25); [r vth]=meshgrid(r,vth); x=r.*cos(vth); y=r.*sin(vth); z=r.^2; mesh(x,y,z) %% 3d plot (improvement 2) clear, clf axes('Pos',[.2 8]) r=linspace(0,1,25); vth=linspace(0,2*pi,25); [r vth]=meshgrid(r,vth); x=r.*cos(vth); y=r.*sin(vth); z=r.^2; surfl(x,y,z+.5), hold on shading interp, colormap copper contour(x,y,z), hold off set(gca, 'XLim',[-1 1], 'YLim',[-1 1], 'ZLim',[-inf inf],'ZTickL',[0:.5:1.5],'ZTickL',[-.5:.5:1], 'Box','on','Color',[1 1]) grid off title('$z=x^2+y^2$') xlabel('$x$') ylabel('$y$') zlabel('$z$') %% movie clear, clf mov=avifile('example.avi') x=linspace(-3,3,25); y=linspace(-3,3,25); [X Y]=meshgrid(x,y); z=peaks(X,Y); surf(x,y,z) shading interp axis([-3 -3 -8 8]) set(gca,'Visible','off') N=25; for I=1:N surf(x,y,sin(2*pi*I/N)*z) shading interp axis([-3 -3 -8 8]) set(gca,'Visible','off') mov=addframe(mov,getframe(gca)); end mov=close(mov); %% matrix multiplication clear N=100; rand('state',1), A=ceil(5*rand(N)); rand('state',2), B=ceil(5*rand(N)); C=zeros(N); tic for I=1:N for J=1:N dmy=0; for K=1:N dmy=dmy+A(I,K)*B(K,J); end C(I,J)=dmy; end end toc tic, C=A*B; toc %% determinants, eigenvalues, etc clear A=[1 0;0 5]; det(A) [V D]=eig(A) A*V D*V %% fourier series clear, clf t=linspace(-.5,.5,250); f=.5*(square(t+.25)-square(t-.25)); h=[]; h=plot(t,f,'r','LineW',1); hold on n=1:15; A=[.5 sin(n*pi/2)./(n*pi/2)]; fn=A*cos(2*[0 n]'*pi*t); h=[h plot(t,fn,'b','LineW',.5)]; hold off set(gca, 'XLim',[-.5 5],'XTick',[-.5 5], 'YLim',[-.5 1.5], 'Box','off') legend(h,{'$f(t)$',['$f_{' num2str(max(n)) '}(t)$']}) xlabel('$t$') set(0,'DefaultTextInterpreter','TeX') function b=num2bin(n) %NUM2BIN Convert floating point number to binary string % NUM2BIN(N) returns the binary representation of the floating point % number N in the form [sign][exponent][mantissa] % % Example % float2bin(single(9.4)) returns % [0][10000000010][0010110011001100110011000000000000000000000000000000] % float2bin(double(9.4)) returns % [0][10000000010][0010110011001100110011001100110011001100110011001101] % % See also DEC2BIN, BIN2DEC, HEX2DEC, BASE2DEC % % Francisco J Beron-Vera $Revision: 1.2 $ $Date: 2006/08/30 16:13:31 $ % Convert to hex string h=sprintf('%bx',n); % Convert to cell array of strings h=cellstr(h'); % Convert hex numerals to decimal numbers h=hex2dec(h); % Convert numeric array to array of binary strings b=dec2bin(h); % Pad with zeros if necessary b=[repmat(num2str(zeros(16,1)),[1 4-size(b,2)]) b]; % Concatenate binary strings b=b'; b=b(:)'; % Output as a string of the form [sign][exponent][mantissa] b=['[' b(1) '][' b(2:12) '][' b(13:end) ']']; function b=double2bin(n) %DOUBLE2BIN Convert double precision floating point to binary string % DOUBLE2BIN(N) returns the binary representation of the double precision % floating point number N in the form [sign][exponent][mantissa] % % The binary representation of F is computed, not converted from its % hexadecimal representation as in NUM2BIN % % See also NUM2BIN, DEC2BIN, BIN2DEC, HEX2DEC, BASE2DEC % % Francisco J Beron-Vera $Revision: 1.3 $ $Date: 2006/08/30 16:19:35 $ % Convert integer part to binary string i=abs(fix(n)); ib=[]; while i~=0 ib=[ib num2str(rem(i,2))]; i=fix(i/2); end if isempty(ib) ib=['0']; end ib=fliplr(ib); % Convert fractional part to binary string f=abs(n-fix(n)); fb=[]; for j=1:53 fb=[fb num2str(fix(f*2))]; f=f*2-fix(f*2); end % Output as a string of the form [sign][exponent][mantissa] if n>0 s='0'; else s='1'; end if length(ib)==1 if str2num(ib)==0 e=[dec2bin(0) dec2bin(1022)]; m=fb(2:end); else e=[dec2bin(0) dec2bin(1023)]; m=fb; end else e=dec2bin(length(ib)+1022); m=[ib(2:end) fb]; end b=['[' s '][' e '][' m(1:52) ']']; function v = longhex(u, digits) %LONGHEX Long hexadecimal format % V=LONGHEX(U,N) takes a positive decimal fraction less than and converts % it to a binary fraction of length digits It then converts it to a IEEE % double precision number but possibly with a longer mantissa U is a vector % of decimal digits in the range [0-9] with the decimal point to the 'left' % of U(1) N is the length in binary digits to save V is a string of hexidecimal % digits of the IEEE double number, but with possibly longer mantiassa % % Algorithm: Knuth, Vol 2, p 319, Method 2a % % % % By: Martin Knapp-Cordes The MathWorks, Inc Date: Apr 2001 - 09 u2 = zeros(size(u,1)+1,1); u2(2:end) = u; b = zeros(digits,1); for i=1:digits u2 = mult2(u2); b(i) = u2(1); u2(1) = 0; end % Always produce an least an IEEE double o = find(b == 1); base = o(1); s = size(b,1); v = dec2hex(1023-base,3); n = max(ceil((s - base) / 4), 13); v2 = zeros(n * 4,1); v2(1:(s - base)) = b((base+1):end); for i=1:4:n*4 v = [ v dec2hex(bin2dec(sprintf('%d',v2(i:i+3))))]; end v = lower(v); % -function u2 = mult2(u) % Multiply by with carry u2 = * u; c = (u2>=10); u2(1:end-1) = u2(1:end-1) + c(2:end) - c(1:end-1) * 10; u2(end) = u2(end) - c(end) * 10; function y=logistic(x,r,anything) %LOGISTIC Logistic map % Y=LOGISTIC(X,R) evaluates R*X*(1-X) % Y=LOGISTIC(X,R,'EXPANDED') evaluates R*X*-R*X*X % % Francisco J Beron-Vera $Revision: 1.0 $ $Date: 2006/09/05 11:52:25 $ if nargineps*abs(x1) k=k+1; if k>25 warning('Tolerance not satisfied.') break end x=f(x0,varargin{:}); plot([x0 x],[x x],'r'), pause plot([x x],[x f(x,varargin{:})],'r'), pause x1=x0; x0=x; end hold off function y=newton(f,dfdx,x,xmin,xmax,varargin) %NEWTON Root finding, Newton method % X=NEWTON(F,DFDX,X0,A,B) attempts to find the solution of F(X)=0 near % X0 where DFDX is the derivative of the scalar function F defined % in the interval [A,B] using Newton method Both F and DFDX are function % handles The geometry of the method is ilustrated and convegence plots % are produced % % X=NEWTON(F,DFDX,X0,A,B,P1,P2, ) passes the parameters P1,P2, to F % % See also BISECTION, SECANT, IQI % % Francisco J Beron-Vera, 2006/09/07 $Revision: 1.1 $ $Date: 2006/09/07 09:00:25 $ % Plot function figure(1), clf X=linspace(xmin,xmax,100); plot(X,f(X,varargin{:}),'b') axis tight, box off, hold on xlabel('$x$', 'FontSize',14, 'Interpreter','LaTeX') ylabel('$y$', 'FontSize',14, 'Interpreter','LaTeX') disp('Press any key to continue.'), pause plot([X(1) X(end)],[0 0],'k'), pause plot([x x],[0 f(x,varargin{:})],'g.:'), pause % Newton loop k=0; y=x+1; Y=[]; while abs(y-x)>eps*abs(x) k=k+1; if k>50 warning('Tolerance not satisfied.') break end y=x; Y=[Y y]; x=x-f(x,varargin{:})/dfdx(x,varargin{:}); plot(X,f(y,varargin{:})+dfdx(y,varargin{:})*(X-y),'r'), pause plot([x x],[0 f(x,varargin{:})],'g.:'), pause end % Convergence analysis figure(2), clf k=0:length(Y)-1; e=abs(Y-Y(end)); subplot(211) plot(k,e) xlabel('$n$', 'FontSize',14, 'Interpreter','LaTeX') ylabel('$e_n:=|x_n-r|$', 'FontSize',14, 'Interpreter','LaTeX') title(['$r=' sprintf('%16.8f',Y(end)) '$'], 'FontSize',14, 'Interpreter','LaTeX') subplot(212) plot(k(2:end-1),e(2:end-1)./e(1:end-2).^2) xlabel('$k$', 'FontSize',14, 'Interpreter','LaTeX') ylabel('$e_n/e_{n-1}^2$', 'FontSize',14, 'Interpreter','LaTeX') function y=newtonsys(f,dfdx,x,varargin) %NEWTONSYS Root finding, Newton method, systems of equations % X=NEWTONSYS(F,DFDX,X0) attempts to find the solution of the nonlinear % system of equations F(X)=0 near X0 where DFDX is the derivative (Jacobian % matrix) of the function F using Newton method % % X=NEWTONSYS(F,DFDX,X0,P1,P2, ) passes the parameters P1,P2, to F % Francisco J Beron-Vera,2006/09/18 y=x+1; k=0; while norm(y-x)>eps*norm(y) k=k+1; if k>50 warning('Tolerance not satisfied.') break end y=x; x=x-dfdx(x,varargin{:})\f(x,varargin{:}); end function r=bisection(f,a,b,varargin) %BISECTION Root finding, bisection method % X=BISECTION(F,A,B) attempts to find the solution of F(X)=0 in the % interval [A,B] such that F(A)*F(B) < % % X=BISECTION(F,A,B,P1,P2, ) passes the parameters P1,P2, to F % % See also NEWTON, SECANT, IQI % % Francisco J Beron-Vera $Revision: 1.1 $ $Date: 2006/09/19 13:28:03 $ if f(a,varargin{:})*f(b,varargin{:})>=0 error('F(A)*F(B) must be negative.') end for k=1:54 r=(a+b)/2; if f(a,varargin{:})*f(r,varargin{:})0 plot(x([2 n+1]),f(x([2 n+1]),varargin{:}),'r'), pause else plot(x([1 n+1]),f(x([1 n+1]),varargin{:}),'r'), pause end end end r=x(n); if strcmp(ConvergenceAnalysis,'yes') % Convergence analysis figure(2), clf x=x(1:n); n=0:n-1; e=abs(x-x(end)); subplot(211) plot(n,e) xlabel('$n$', 'FontSize',14, 'Interpreter','LaTeX') ylabel('$e_n:=|x_n-r|$', 'FontSize',14, 'Interpreter','LaTeX') title(['$r=' sprintf('%16.8f',x(end)) '$'], 'FontSize',14, 'Interpreter','LaTeX') subplot(212) plot(n(2:end-1),e(2:end-1)./e(1:end-2).^1.62), hold on h=plot(n([2 end-1]),repmat(.5*d2fdx2(r)/dfdx(r),[1 2]),'r:'); hold off legend(h,{['$f''''(r)/2f''(r)\approx' num2str(.5*d2fdx2(r)/dfdx(r)) '$']}, 'FontSize',14, 'Interpreter','LaTeX') legend boxoff xlabel('$k$', 'FontSize',14, 'Interpreter','LaTeX') ylabel('$e_n/e_{n-1}^{1.62}$', 'FontSize',14, 'Interpreter','LaTeX') end function r=iqi(f,x1,x2,x3,varargin) %IQI Root finding, inverse quadratic interpolation method % X=IQI(F,X1,X2,X3) attempts to find the solution of F(X)=0 given initial % guesses X1, X2 and X3 using the inverse quadratic interpolation method % F is a function handle % % X=IQI(F,X1,X2,X3,P1,P2, ) passes the parameters P1,P2, to F n = n+1; if n > N warning('Tolerance not satisfied.') break end y(n+1) = y(n)-E(n)*(y(n)-y(n-1))/(E(n)-E(n-1)); if length(xspan) == [x z] = ode45(odefun,x,[A;y(n+1)],options,varargin{:}); else z = ode4(odefun,x,[A;y(n+1)],varargin{:}); end u(:,n+1) = z(:,1); E(n+1) = z(end,1)-B; end if length(xspan)==2 varargout{1} = x; else varargout{1} = x; end varargout{2} = u(:,1:n); if nargout > varargout{3} = y(1:n); varargout{4} = E(1:n); end % Solves numerically the one-dimensional diffusion equation, % % u_t = u_{xx} (x,t) \in [0,1]^2 % u(x,0) = \sin \pi x x \in [0,1] % u(0,t) = t \in [0,1] % u(1,t) = t \in [0,1] % % whose exact solution is % % u(x,t) = \sin \pi x \exp(-\pi^2 t), % % using forward-time difference, backward-time difference, and % Crank-Nicolson methods % Francisco J Beron-Vera, 2006/15/05 %% FORWARD-TIME DIFFERENCE (EXPLICIT) % % U(I,J) = u((I-1)*Dx,(J-1)*Dt), I = 1, ,Nx+1, J = 1, ,Nt+1 % (U(I,J+1) - U(I,J))/Dt = (U(I+1,J) - 2*U(I,J) + U(I-1,J))/Dx^2 % U(I,1) = sin(pi*(I-1)*Dx), U(1,J) = 0, U(Nx+1,J) = % s = Dt/Dx^2, Nx = 6: % |U(2,J+1)| |1-2*s s 0 | |U(2,J)| |s*U(1,J)=0| % |U(3,J+1)| | s 1-2*s s 0 | |U(3,J)| | | % |U(4,J+1)| = | s 1-2*s s | |U(4,J)| + | | % |U(5,J+1)| | 0 s 1-2*s s | |U(5,J)| | | % |U(6,J+1)| | 0 s 1-2*s| |U(6,J)| |s*U(7,J)=0| % Order of local truncation error: Dt + Dx^2 % Conditionally stable (s < 1/2) clear Nx = 50; Dx = 1/Nx; x = (0:Nx)'*Dx; Dt = 49/Nx^2; Nt = fix(1/Dt); t = (0:Nt)'*Dt; s = Dt/Dx^2; T = diag(repmat(s,Nx-2,1),1) + (1-2*s)*eye(Nx-1) + diag(repmat(s,Nx-2,1),-1); U = zeros(Nx-1,Nt+1); U(:,1) = sin(pi*x(2:end-1)); for J = 1:Nt-1 U(:,J+1) = T*U(:,J); end U = [zeros(1,Nt+1); U; zeros(1,Nt+1)]; figure(1), clf subplot(221) plot(x,U(:,[1 fix(rand(1,25)*Nt)+1]),'b') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x)$','FontS',14,'Interp','latex') text(1.25,1.15,['Forward-Time Difference ($N_x = ' num2str(Nx) '$, $N_t = ' num2str(Nt) '$)'], 'Hor','cen','FontS',14,'Interp','latex') subplot(222) surf(t,x,U), shading flat set(gca,'YDir','reverse') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$x$','FontS',14,'Interp','latex') zlabel('$u(x,t)$','FontS',14,'Interp','latex') subplot(223) [X T] = meshgrid(x,t); Ux05 = interp2(X,T,U',repmat(.5,size(t)),t); plot(t,Ux05,'bo'), hold on h = plot(t,sin(pi*.5)*exp(-pi^2*t),'r'); h =legend(h,'exact'); set(h,'FontSize',14,'Interp','latex') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$u(x=0.5,t)$','FontS',14,'Interp','latex') subplot(224) Ut05 = interp2(X,T,U',x,repmat(.5,size(x))); plot(x,Ut05,'bo',x,sin(pi*x)*exp(-pi^2*.5),'r') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x,t=0.5)$','FontS',14,'Interp','latex') %% BACKWARD-TIME DIFFERENCE (IMPLICIT) % % U(I,J) = u((I-1)*Dx,(J-1)*Dt), I = 1, ,Nx+1, J = 1, ,Nt+1 % (U(I,J) - U(I,J-1))/Dt = (U(I+1,J) - 2*U(I,J) + U(I-1,J))/Dx^2 % U(I,1) = sin(pi*(I-1)*Dx), U(1,J) = 0, U(Nx+1,J) = % s = Dt/Dx^2, Nx = 6: % |1+2*s -s 0 | |U(2,J)| |U(2,J-1)| |s*U(1,J)=0| % | -s 1+2*s -s 0 | |U(3,J)| |U(3,J-1)| | | % | -s 1+2*s -s | |U(4,J)| = |U(4,J-1)| + | | % | 0 -s 1+2*s -s | |U(5,J)| |U(5,J-1)| | | % | 0 -s 1+2*s| |U(6,J)| |U(6,J-1)| |s*U(7,J)=0| % Order of local truncation error: Dt + Dx^2 % Unconditionally stable clear Nx = 50; Dx = 1/Nx; x = (0:Nx)'*Dx; Nt = 1500; Dt = 1/Nt; t = (0:Nt)'*Dt; s = Dt/Dx^2; T = diag(repmat(-s,Nx-2,1),1) + (1+2*s)*eye(Nx-1) + diag(repmat(-s,Nx-2,1),1); U = zeros(Nx-1,Nt+1); U(:,1) = sin(pi*x(2:end-1)); for J = 2:Nt+1 U(:,J) = T\U(:,J-1); end U = [zeros(1,Nt+1); U; zeros(1,Nt+1)]; figure(2), clf subplot(221) plot(x,U(:,[1 fix(rand(1,25)*Nt)+1]),'b') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x)$','FontS',14,'Interp','latex') text(1.25,1.15,['Backward-Time Difference ($N_x = ' num2str(Nx) '$, $N_t = ' num2str(Nt) '$)'], 'Hor','cen','FontS',14,'Interp','latex') subplot(222) surf(t,x,U), shading flat set(gca,'YDir','reverse') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$x$','FontS',14,'Interp','latex') zlabel('$u(x,t)$','FontS',14,'Interp','latex') subplot(223) [X T] = meshgrid(x,t); Ux05 = interp2(X,T,U',repmat(.5,size(t)),t); plot(t,Ux05,'bo'), hold on h = plot(t,sin(pi*.5)*exp(-pi^2*t),'r'); h =legend(h,'exact'); set(h,'FontSize',14,'Interp','latex') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$u(x=0.5,t)$','FontS',14,'Interp','latex') subplot(224) Ut05 = interp2(X,T,U',x,repmat(.5,size(x))); plot(x,Ut05,'bo',x,sin(pi*x)*exp(-pi^2*.5),'r') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x,t=0.5)$','FontS',14,'Interp','latex') %% CRANK-NICOLSON (IMPLICIT) % % U(I,J) = u((I-1)*Dx,(J-1)*Dt), I = 1, ,Nx+1, J = 1, ,Nt+1 % (U(I,J) - U(I,J-1))/Dt = (U(I+1,J) - 2*U(I,J) + U(I-1,J) + U(I+1,J-1) 2*U(I,J-1) + U(I-1,J-1))/(2*Dx^2) % U(I,1) = sin(pi*(I-1)*Dx), U(1,J) = 0, U(Nx+1,J) = % s = Dt/Dx^2, Nx = 6: % |2+2*s -s 0 | |U(2,J)| |2-2*s s 0 | |U(2,J1)| |s*(U(1,J-1)+U(1,J))=0| % | -s 2+2*s -s 0 | |U(3,J)| | s 2-2*s s 0 | 1)| | | % | -s 2+2*s -s | |U(4,J)| = | s 2-2*s -s | 1)| + | | % | 0 -s 2+2*s -s | |U(5,J)| | 0 s 2-2*s s | 1)| | | % | 0 -s 2+2*s| |U(6,J)| | 0 s 2-2*s| 1)| |s*(U(1,J-1)+U(1,J))=0| % Order of local truncation error: Dt^2 + Dx^2 % Unconditionally stable |U(3,J|U(4,J|U(5,J|U(6,J- clear Nx = 50; Dx = 1/Nx; x = (0:Nx)'*Dx; Nt = 100; Dt = 1/Nt; t = (0:Nt)'*Dt; s = Dt/Dx^2; T1 = diag(repmat(-s,Nx-2,1),1) + (2+2*s)*eye(Nx-1) + diag(repmat(-s,Nx-2,1),1); T2 = diag(repmat(s,Nx-2,1),1) + (2-2*s)*eye(Nx-1) + diag(repmat(s,Nx-2,1),-1); U = zeros(Nx-1,Nt+1); U(:,1) = sin(pi*x(2:end-1)); for J = 2:Nt+1 U(:,J) = T1\(T2*U(:,J-1)); end U = [zeros(1,Nt+1); U; zeros(1,Nt+1)]; figure(3), clf subplot(221) plot(x,U(:,[1 fix(rand(1,25)*Nt)+1]),'b') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x)$','FontS',14,'Interp','latex') text(1.25,1.15,['Crank Nicolson ($N_x = ' num2str(Nx) '$, $N_t = ' num2str(Nt) '$)'], 'Hor','cen','FontS',14,'Interp','latex') subplot(222) surf(t,x,U), shading flat set(gca,'YDir','reverse') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$x$','FontS',14,'Interp','latex') zlabel('$u(x,t)$','FontS',14,'Interp','latex') subplot(223) [X T] = meshgrid(x,t); Ux05 = interp2(X,T,U',repmat(.5,size(t)),t); plot(t,Ux05,'bo'), hold on h = plot(t,sin(pi*.5)*exp(-pi^2*t),'r'); h =legend(h,'exact'); set(h,'FontSize',14,'Interp','latex') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$u(x=0.5,t)$','FontS',14,'Interp','latex') subplot(224) Ut05 = interp2(X,T,U',x,repmat(.5,size(x))); plot(x,Ut05,'bo',x,sin(pi*x)*exp(-pi^2*.5),'r') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x,t=0.5)$','FontS',14,'Interp','latex') % Solves numerically a two-dimensional diffusion equation, using forward% time difference, backward-time difference, Crank-Nicolson, or Peaceman% Rachford method % Francisco J Beron-Vera, 2006/15/05 clear disp('Solves numerically the two-dimensional diffusion equation:') disp(' ') disp(' u_t = u_{xx} +u_{yy} (x,y,t) \in [0,1]^2 \times [0,0.1],') disp(' u(x,y,0) = \sin \pi x \sin \pi y (x,y) \in [0,1]^2,') disp(' u(0,y,t) = (y,t) \in [0,1] \times [0,0.1],') disp(' u(1,y,t) = (y,t) \in [0,1] \times [0,0.1],') disp(' u(x,0,t) = (x,t) \in [0,1] \times [0,0.1],') disp(' u(x,1,t) = (x,t) \in [0,1] \times [0,0.1],') disp(' ') disp('whose exact solution is') disp(' ') disp(' u(x,y,t) = \sin \pi x \sin \pi y \exp(-2\pi^2 t).') disp(' ') disp('Available methods:') disp(' ') disp('(1) Forward-time difference : U(t+Dt) = Pfd*U(t) (N.B Dx^2 + Dy^2 > 4*Dt)') disp('(2) Backward-time difference: Pbd*U(t+Dt) = U(t)') disp('(3) Crank-Nicolson : Pcn1*U(t+Dt) = Pcn2*U(t)') disp('(4) Peaceman-Rachford : Ppr1x*U(t+Dt) = Ppr2y*U(t)') disp(' Ppr1y*U(t+Dt) = Ppr2x*U(t+Dt)') disp(' ') method = input('Your method choice: '); Nx = input('Number of grid points in x direction: ') - 1; Ny = input('Number of grid points in y direction: ') - 1; Nt = input('Number of time steps = '); u = @(x,y,t) sin(pi*x).*sin(pi*y)*exp(-2*pi^2*t); Dx = 1/Nx; x = (0:Nx)'*Dx; Dy = 1/Ny; y = (0:Ny)'*Dy; Dt = 1/Nt; t = (0:Nt)'*Dt; sx = Dt/Dx^2; sy = Dt/Dy^2; nx ny Sx Sy Z E Ex Ey = = = = = = = = d2dx2 d2dy2 Nx-1; Ny-1; repmat(sx,nx,1); repmat(sy,ny,1); spalloc(nx*ny,1,nx*ny); speye(nx*ny); speye(nx); speye(ny); = spdiags([Sx -2*Sx Sx],-1:1,nx,nx); = spdiags([Sy -2*Sy Sy],-1:1,ny,ny); nabla2 = kron(d2dy2,Ex) + kron(Ey,d2dx2); nabla2x = spdiags([Z diag(nabla2)+2*sy Z],[-nx nx],nabla2); nabla2y = spdiags([Z diag(nabla2)+2*sx Z],-1:1,nabla2); if method == Pfd = E + nabla2; elseif method == Pbd = E - nabla2; elseif method == Pcn1 = E - nabla2/2; Pcn2 = E + nabla2/2; else Ppr1x = E - nabla2x/2; Ppr2x = E + nabla2x/2; Ppr1y = E - nabla2y/2; Ppr2y = E + nabla2y/2; end % Initializing U [x y] = ndgrid(x,y); UdU = u(x,y,0); subplot(221) pcolor(x,y,UdU) axis([0 1]) shading flat, caxis([0 1]) set(gca,'XTick',[0 1],'YTick',[0 1]) xlabel('x'), ylabel('y'), title('NUMERIC') text(1.15,1.125,'t = 0','Hor','cen') subplot(222) pcolor(x,y,UdU) axis([0 1]) shading flat, caxis([0 1]) h = colorbar('SouthOutside'); set(h,'Pos',[.3 45 025]) set(get(h,'XLabel'),'Str','u') set(gca,'XTick',[0 1],'YTick',[0 1]) xlabel('x'), title('EXACT') drawnow U = UdU; U([1 end],:) = []; U(:,[1 end]) = []; U = U(:); % Time stepping U for J = 2:Nt+1 if method == U = Pfd*U; elseif method == U = Pbd\U; elseif method == U = Pcn1\(Pcn2*U); else U = Ppr1x\(Ppr2y*U); U = Ppr1y\(Ppr2x*U); end subplot(221) UdU(2:end-1,2:end-1) = reshape(U,[nx ny]); pcolor(x,y,UdU) axis([0 1]) shading flat, caxis([0 1]) set(gca,'XTick',[0 1],'YTick',[0 1]) xlabel('x'), ylabel('y'), title('NUMERIC') text(1.15,1.125,['t = ' num2str(t(J))],'Hor','cen') subplot(222) pcolor(x,y,u(x,y,t(J))) axis([0 1]) shading flat, caxis([0 1]) h = colorbar('SouthOutside'); set(h,'Pos',[.3 45 025]) set(get(h,'XLabel'),'Str','u') set(gca,'XTick',[0 1],'YTick',[0 1]) xlabel('x'), title('EXACT') drawnow end % Solves numerically the one-dimensional wave equation, % % u_t = u_{xx} (x,t) \in [0,1]^2 % u(x,0) = \sin 2\pi x x \in [0,1] % u_t(x,0) = x \in [0,1] % u(0,t) = t \in [0,1] % u(1,t) = t \in [0,1] % % whose exact solution is % % u(x,t) = \sin 2\pi x \cos 2\pi t), % % using explicit and implicit methods % Francisco J Beron-Vera, 2006/15/05 %% EXPLICIT METHOD % U(I,J) = u((I-1)*Dx,(J-1)*Dt), I = 1, ,Nx+1, J = 1, ,Nt+1 % (U(I,J+1) - 2*U(I,J) + U(I,J-1)/Dt^2 = (U(I+1,J) - 2*U(I,J) + U(I-1,J))/Dx^2 % U(I,1) = sin(2*pi*(I-1)*Dx), U(I,2) = U(I,2), U(1,J) = 0, U(Nx+1,J) = % Order of local truncation error: Dt^2 + Dx^2 % Conditionally stable: Dx^2/Dt^2 > (Courant-Friedrichs-Lewy) clear u = @(x,t) sin(2*pi*x).*cos(2*pi*t); Nx = 50; Dx = 1/Nx; x = (0:Nx)'*Dx; Dt = 25/Nx; Nt = fix(1/Dt); t = (0:Nt)'*Dt; s = Dt^2/Dx^2; S = repmat(s,Nx-1,1); T = spdiags([S 2-2*S S],-1:1,Nx-1,Nx-1); U = zeros(Nx-1,Nt+1); U(:,1) = u(x(2:end-1),0); U(:,2) = U(:,1); for J = 2:Nt U(:,J+1) = T*U(:,J) - U(:,J-1); end U = [zeros(1,Nt+1); U; zeros(1,Nt+1)]; figure(1), clf subplot(221) plot(x,U(:,[1 fix(rand(1,15)*Nt)+1]),'b') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x)$','FontS',14,'Interp','latex') text(1.25,1.2,['Explicit Method ($N_x = ' num2str(Nx) '$, $N_t = ' num2str(Nt) '$)'], 'Hor','cen','FontS',14,'Interp','latex') subplot(222) surf(t,x,U), shading flat set(gca,'YDir','reverse') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$x$','FontS',14,'Interp','latex') zlabel('$u(x,t)$','FontS',14,'Interp','latex') subplot(223) [X T] = meshgrid(x,t); Ux025 = interp2(X,T,U',repmat(.25,size(t)),t); plot(t,Ux025,'bo'), hold on h = plot(t,u(.25,t),'r'); h =legend(h,'exact'); set(h,'FontSize',14,'Interp','latex') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$u(x=0.25,t)$','FontS',14,'Interp','latex') subplot(224) Ut05 = interp2(X,T,U',x,repmat(.5,size(x))); plot(x,Ut05,'bo',x,u(x,.5),'r') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x,t=0.5)$','FontS',14,'Interp','latex') %% IMPLICIT METHOD % U(I,J) = u((I-1)*Dx,(J-1)*Dt), I = 1, ,Nx+1, J = 1, ,Nt+1 % (U(I,J+1) - 2*U(I,J) + U(I,J-1)/Dt^2 = (U(I+1,J+1) - 2*U(I,J+1) + U(I1,J+1))/(2*Dx^2) + % (U(I+1,J-1) - 2*U(I,J-1) + U(I-1,J1))/(2*Dx^2) % U(I,1) = sin(2*pi*(I-1)*Dx), U(I,2) = U(I,2), U(1,J) = 0, U(Nx+1,J) = % Order of local truncation error: Dt^2 + Dx^2 % Unconditionally stable clear u = @(x,t) sin(2*pi*x).*cos(2*pi*t); Nx = 50; Dx = 1/Nx; x = (0:Nx)'*Dx; Nt = 50; Dt = 1/Nt; t = (0:Nt)'*Dt; s = Dt^2/Dx^2; S = repmat(s,Nx-1,1); T = spdiags([-.5*S 1+S -.5*S],-1:1,Nx-1,Nx-1); U = zeros(Nx-1,Nt+1); U(:,1) = u(x(2:end-1),0); U(:,2) = U(:,1); for J = 2:Nt U(:,J+1) = T\(2*U(:,J)-T*U(:,J-1)); end U = [zeros(1,Nt+1); U; zeros(1,Nt+1)]; figure(2), clf subplot(221) plot(x,U(:,[1 fix(rand(1,15)*Nt)+1]),'b') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x)$','FontS',14,'Interp','latex') text(1.25,1.2,['Implicit Method ($N_x = ' num2str(Nx) '$, $N_t = ' num2str(Nt) '$)'], 'Hor','cen','FontS',14,'Interp','latex') subplot(222) surf(t,x,U), shading flat set(gca,'YDir','reverse') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$x$','FontS',14,'Interp','latex') zlabel('$u(x,t)$','FontS',14,'Interp','latex') subplot(223) [X T] = meshgrid(x,t); Ux025 = interp2(X,T,U',repmat(.25,size(t)),t); plot(t,Ux025,'bo'), hold on h = plot(t,u(.25,t),'r'); h =legend(h,'exact'); set(h,'FontSize',14,'Interp','latex') xlabel('$t$','FontS',14,'Interp','latex') ylabel('$u(x=0.25,t)$','FontS',14,'Interp','latex') subplot(224) Ut05 = interp2(X,T,U',x,repmat(.5,size(x))); plot(x,Ut05,'bo',x,u(x,.5),'r') xlabel('$x$','FontS',14,'Interp','latex') ylabel('$u(x,t=0.5)$','FontS',14,'Interp','latex') % Solves numerically a two-dimensional wave equation, using explicit or % implicit methods % Francisco J Beron-Vera, 2006/15/05 clear disp('Solves numerically the two-dimensional wave equation:') disp(' ') disp(' u_{tt} = u_{xx} +u_{yy} (x,y,t) \in [0,1] \times [0,5],') disp(' u(x,y,0) = \sin 2\pi x \sin 2\pi y (x,y) \in [0,1]^2,') disp(' u_t(x,y,0) = (x,y) \in [0,1]^2,') disp(' u(0,y,t) = (y,t) \in [0,1] \times [0,5],') disp(' u(1,y,t) = (y,t) \in [0,1] \times [0,5],') disp(' u(x,0,t) = (x,t) \in [0,1] \times [0,5],') disp(' u(x,1,t) = (x,t) \in [0,1] \times [0,5],') disp(' ') disp('whose exact solution is') disp(' ') disp(' u(x,y,t) = \sin 2\pi x \sin 2\pi y \cos(2\pi t).') disp(' ') disp('Available methods:') disp(' ') disp('(1) Explicit: U(t+Dt) = T*U(t) - U(t-Dt) (N.B (Dx^2 + Dy^2)/Dt^2 > 1)') disp('(2) Implicit: T*U(t+Dt) = 2*U(t) - T*U(t-Dt)') disp(' ') method = input('Your method choice: '); Nx = input('Number of grid points in x direction: ') - 1; Ny = input('Number of grid points in y direction: ') - 1; Nt = input('Number of time steps = '); u = @(x,y,t) sin(2*pi*x).*sin(2*pi*y)*cos(2*pi*t); Dx = 1/Nx; x = (0:Nx)'*Dx; Dy = 1/Ny; y = (0:Ny)'*Dy; Dt = 5/Nt; t = (0:Nt)'*Dt; sx = Dt^2/Dx^2; sy = Dt^2/Dy^2; nx ny Sx Sy E Ex Ey = = = = = = = Nx-1; Ny-1; repmat(sx,nx,1); repmat(sy,ny,1); speye(nx*ny); speye(nx); speye(ny); d2dx2 = spdiags([Sx -2*Sx Sx],-1:1,nx,nx); d2dy2 = spdiags([Sy -2*Sy Sy],-1:1,ny,ny); nabla2 = kron(d2dy2,Ex) + kron(Ey,d2dx2); if method == T = 2*E + nabla2/2; else T = E - nabla2/4; end % Initializing U [x y] = ndgrid(x,y); UdU = u(x,y,0); subplot(221) surf(x,y,UdU), shading flat axis([0 1 -1 1]) xlabel('x'), ylabel('y'), zlabel('u'), title('NUMERIC') text(2.05,1,.5,'t = 0','Hor','cen') subplot(222) surf(x,y,UdU), shading flat axis([0 1 -1 1]) xlabel('x'), ylabel('y'), zlabel('u'), title('EXACT') drawnow Uold = UdU; Uold([1 end],:) = []; Uold(:,[1 end]) = []; Uold = Uold(:); U = Uold; % Time stepping U for J = 2:Nt+1 if method == Unew = T*U - Uold; else Unew = T\(2*U - T*Uold); end Uold = U; U = Unew; subplot(221) UdU(2:end-1,2:end-1) = reshape(Unew,[nx ny]); surf(x,y,UdU), shading flat axis([0 1 -1 1]) xlabel('x'), ylabel('y'), zlabel('u'),title('NUMERIC') text(2.05,1,.5,['t = ' num2str(t(J))],'Hor','cen') subplot(222) surf(x,y,u(x,y,t(J))), shading flat axis([0 1 -1 1]) xlabel('x'), ylabel('y'), zlabel('u'), title('EXACT') drawnow end % Solves the Poisson equation with Dirichlet boundary conditions in a unit % square given by % % u_{xx} + u_{yy} = \exp xy (x,y) \in [0,1]^2, % u(0,y) = y (y - 1) y \in [0,1], % u(1,y) = - y ((y-1)^4) y \in [0,1], % u(x,0) = 0.5 sin 6\pi x x \in [0,1], % u(x,1) = sin 2\pi x x \in [0,1] % % Compares Gaussian eleimination (Matlab's backslash division)and fast sine % transfom methods clear, clf x = linspace(0,1,128+1); y = linspace(0,1,64+1)'; Dx = min(diff(x)); Dy = min(diff(y)); [X Y] = meshgrid(x,y); f = exp(X.*Y); f(:,1) = 5.*y.*(y-1); f(:,end) = -y.*((y-1).^4); f(1,:) = 5.*sin(6*pi.*x); f(end,:) = sin(pi*2.*x); % % % % east west south north subplot(221) tic, u = ps2d(f,Dx,Dy); et = toc; mesh(X,Y,u) xlabel('x'), ylabel('y'), zlabel('u') title({'Gaussian elimination';['(elapsed time = ' num2str(et) ')']}) subplot(222) tic, u = fps2d(f,Dx,Dy); et = toc; mesh(X,Y,u) xlabel('x'), ylabel('y'), zlabel('u') title({'fast sine transform';['(elapsed time = ' num2str(et) ')']}) function U = ps2d(F,Dx,Dy) %PS2D 2D Poisson equation solver % U = FPS2D(F,DX,DY) solves the Poisson equation in two-space dimensions % with source term F(X,Y) defined on a rectangular domain and Dirichlet % boundary conditions The boundary terms are specified in the perimeter % of the NY x NX matrix F generated using meshgrid Thus EAST, WEST, % SOUTH, and NORTH boundary conditions are to be specified, respectively, % in F(:,1) F(:,END), F(1,:), and F(END,:) Meshgrid widths in X and Y % directions are specified in DX and DY, respectively Inversion of the % five-point discrete Laplacian is done by Gaussian elemination as % implemented in Matlab's backslash division % % See also FPS2D, HS2D, FHS2D % % Francisco J Beron-Vera, 2006/11/15 $Revision: 1.1 $ $Date: 2006/11/22 22:01:20 $ % Get dimensions Nx = size(F,2)-2; Ny = size(F,1)-2; % Five-point discrete Laplacian DX = repmat(Dx,Nx,1); DY = repmat(Dy,Ny,1); d2dx2 = spdiags([1./DX.^2 -2./DX.^2 1./DX.^2],-1:1,Nx,Nx); d2dy2 = spdiags([1./DY.^2 -2./DY.^2 1./DY.^2],-1:1,Ny,Ny); nab2 = kron(d2dx2,speye(Ny)) + kron(speye(Nx),d2dy2); % Fold boundary into source term U = F; U(2:end-1,2:end-1) = 0; U(2:end-1,2:end-1) = 0; F = F - 4*del2(U,Dx,Dy); F([1 end],:) = []; F(:,[1 end]) = []; F = F(:); % Solve nabla2*U = F by Gaussian elimination U(2:end-1,2:end-1) = reshape(nab2\F,Ny,Nx); function U = fps2d(F,Dx,Dy) %FPS2D Fast 2D Poisson equation solver % U = FPS2D(F,DX,DY) solves the Poisson equation in two-space dimensions % with source term F(X,Y) defined on a rectangular domain and Dirichlet % boundary conditions The boundary terms are specified in the perimeter % of the NY x NX matrix F generated using meshgrid Thus EAST, WEST, % SOUTH, and NORTH boundary conditions are to be specified, respectively, % in F(:,1) F(:,END), F(1,:), and F(END,:) Meshgrid widths in X and Y % directions are specified in DX and DY, respectively Inversion of the % discrete five-point Laplacian is perfomed using the fast sine transform % For best performance speed NX - and NY - should be chosen as powers % of % % See also PS2D, HS2D, FHS2D % % Francisco J Beron-Vera, 2006/11/15 $Revision: 1.1 $ $Date: 2006/11/22 21:11:20 $ % Get dimensions Nx = size(F,2)-2; Ny = size(F,1)-2; % Fold boundary into source term U = F; U(2:end-1,2:end-1) = 0; U(2:end-1,2:end-1) = 0; F = F - 4*del2(U,Dx,Dy); F([1 end],:) = []; F(:,[1 end]) = []; % Eigenvalues of discrete Laplacian (nab2) [nx ny] = meshgrid(1:Nx,1:Ny); L = 2/Dx^2*(cos(nx*pi/(Nx+1))-1) + 2/Dy^2*(cos(ny*pi/(Ny+1))-1); % Solve nab2*U = F using fast sine transform U(2:end-1,2:end-1) = idst(idst(dst(dst(F)')'./L)')'; function U = hs2d(F,k,Dx,Dy) %HPS2D 2D Helmholtz equation solver % U = FHS2D(F,K,DX,DY) solves the Helmholtz equation in two-space % dimensions with source term F(X,Y) defined on a rectangular domain % assuming Dirichlet boundary conditions and for constant wavenumber K % The boundary terms are specified in the perimeter of the NY x NX matrix % F generated using meshgrid Thus EAST, WEST, SOUTH, and NORTH boundary % conditions are to be specified, respectively, in F(:,1) F(:,END), F(1,:), % and F(END,:) Meshgrid widths in X and Y directions are specified in DX % and DY, respectively Inversion of the five-point discrete Laplacian is % done by Gaussian elemination as implemented in Matlab's backslash division % % See also PS2D, FPS2D, FHS2D % Francisco J Beron-Vera, 2006/11/22 % Get dimensions Nx = size(F,2)-2; Ny = size(F,1)-2; % Five-point discrete Laplacian DX = repmat(Dx,Nx,1); DY = repmat(Dy,Ny,1); d2dx2 = spdiags([1./DX.^2 -2./DX.^2 1./DX.^2],-1:1,Nx,Nx); d2dy2 = spdiags([1./DY.^2 -2./DY.^2 1./DY.^2],-1:1,Ny,Ny); nab2 = kron(d2dx2,speye(Ny)) + kron(speye(Nx),d2dy2); % Fold boundary into source term U = F; U(2:end-1,2:end-1) = 0; U(2:end-1,2:end-1) = 0; F = F - 4*del2(U,Dx,Dy); F([1 end],:) = []; F(:,[1 end]) = []; F = F(:); % Solve nabla2*U + k^2*U = F by Gaussian elimination k2 = spdiags(repmat(k^2,Nx*Ny,1),0,Nx*Ny,Nx*Ny); U(2:end-1,2:end-1) = reshape((nab2+k2)\F,Ny,Nx); function U = fhs2d(F,k,Dx,Dy) %FHS2D Fast 2D Helmholtz equation solver % U = FHS2D(F,K,DX,DY) solves the Helmholtz equation in two-space % dimensions with source term F(X,Y) defined on a rectangular domain % assuming Dirichlet boundary conditions and for constant wavenumber K % The boundary terms are specified in the perimeter of the NY x NX matrix % F generated using meshgrid Thus EAST, WEST, SOUTH, and NORTH boundary % conditions are to be specified, respectively, in F(:,1) F(:,END), F(1,:), % and F(END,:) Meshgrid widths in X and Y directions are specified in DX % and DY, respectively Inversion of the discrete five-point Laplacian is % perfomed using the fast sine transform For best performance speed % NX - and NY - should be chosen as powers of % % See also HS2D, PS2D, FPS2D % % Francisco J Beron-Vera, 2006/11/15 $Revision: 1.1 $ $Date: 2006/11/22 21:11:20 $ % Get dimensions Nx = size(F,2)-2; Ny = size(F,1)-2; % Fold boundary into source term U = F; U(2:end-1,2:end-1) = 0; U(2:end-1,2:end-1) = 0; F = F - 4*del2(U,Dx,Dy); F([1 end],:) = []; F(:,[1 end]) = []; % Eigenvalues of discrete Laplacian (nab2) [nx ny] = meshgrid(1:Nx,1:Ny); L = 2/Dx^2*(cos(nx*pi/(Nx+1))-1) + 2/Dy^2*(cos(ny*pi/(Ny+1))-1); % Solve nab2*U + k^2*U = F using fast sine transform U(2:end-1,2:end-1) = idst(idst(dst(dst(F)')'./(L+k^2))')'; ... set(0,'DefaultTextInterpreter','TeX') function b=num2bin(n) %NUM2BIN Convert floating point number to binary string % NUM2BIN(N) returns the binary representation of the floating point % number N in the... b=double2bin(n) %DOUBLE2BIN Convert double precision floating point to binary string % DOUBLE2BIN(N) returns the binary representation of the double precision % floating point number N in the form... r=iqi(f,x1,x2,x3,varargin) %IQI Root finding, inverse quadratic interpolation method % X=IQI(F,X1,X2,X3) attempts to find the solution of F(X)=0 given initial % guesses X1, X2 and X3 using the inverse quadratic interpolation