# Partial differential equations

## Wave equation - explicit finite differences

**Ilustration, how stability of the explicit method depends on the step-length.**

Save the following program with an example to a file name.m and run it in Matlab:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % explicit finite differences for wave equation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all c = 2; % wave speed h = 0.2; % step in space a = -1; b = 1; % interval of solution (in space) xh = a:h:b; % division of the interval with step h n = length(xh); % the number of nodes f0 = 2*xh(2:n-1); % right hand side: f0(x,t) = 2*x U0 = 2*xh.^2; % initial condition: u0 = 2*x*x dU0 = (ones(1,n) - xh).*sin(pi*xh/2); % initial condition: du0/dx hold off plot(xh,U0,'g') % graph of initial solution hold on tau = 0.08; % step in time sig = c*tau/h; % coefficient sigma used at finite diff. method s = sig^2; % all formulas use sigma squared only % the first time level computed from initial conditions: U1(2:n-1) = U0(2:n-1) + tau * dU0(2:n-1); U1(1) = 2*exp(-tau); % b.c. at left - time dependent U1(n) = 2; % b.c. at right - constant plot(xh,U1) % cycle for time levels disp('Press a Space tab for the next time level ... '); for k = 2:8 U2(2:n-1) = s*(U1(1:n-2)+U1(3:n))+2*(1-s)*U1(2:n-1)-U0(2:n-1)+tau^2*f0; U2(1) = 2*exp(-k*tau); % b.c. at left U2(n) = 2; % b.c. at right pause plot(xh,U2) U0 = U1; U1 = U2; end disp('The end.'); disp(''); % change the time step and compute once again disp('Now the same with larger time-step:'); disp('Press a Space tab for the next time level ... '); U0 = 2*xh.^2; % initial condition: u0 = 2*x*x tau = 0.16; % larger time-step sig = c*tau/h; % recompute sigma s = sig^2; % the first time level computed from initial conditions: U1(2:n-1) = U0(2:n-1) + tau * dU0(2:n-1); U1(1) = 2*exp(-tau); % b.c. at left - time dependent U1(n) = 2; % b.c. at right - constant plot(xh,U1) for k = 2:4 U2(2:n-1) = s*(U1(1:n-2)+U1(3:n))+2*(1-s)*U1(2:n-1)-U0(2:n-1)+tau^2*f0; U2(1) = 2*exp(-k*tau); % b.c. at left U2(n) = 2; % b.c. at right pause plot(xh,U2,'r') U0 = U1; U1 = U2; end disp('The end.'); %%%%%%%%%%%%%%%%%% % end of program %%%%%%%%%%%%%%%%%%