Notes for exercises

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NOTES FOR EXERCICES M&N METHODS – NUMERICAL PART MATLAB EXERCICES Order of accuracy: 𝑝=log!&"##(%,'()!"##*"#,$%#+!&'' , 𝑒𝑟𝑟,=|𝑢,−𝑢"-./((𝑥,)| i=1,…,M-1 are NOT the nodes but the halvings of the discretization step h for j = 1:M [t, u] = method(t0, tf, u0, dt, lambda,…); e(j) = max( abs(u – u_ex) ); dt = dt/2; end p = log2( e(1:end-1) ./ e(2:end) ); Transport eq.: • FE/C unew(2:(Nh-1)) = u(2:Nh-1) - CFL/2*( u(3:Nh) - u(1:Nh-2) ) % Boundary condition unew(1) = Uinflow; % Inflow unew(Nh) = (1-CFL)*u(Nh) + CFL*u(Nh-1); % Outflow (UW) • BE/C A = speye(Nh-1) + spdiags([-e, e], [-1, 1], Nh-1, Nh-1); for n = 1:Nt-1 t = t+dt; F(2:Nh-1) = u(3:Nh); F(1) = u(2) + 0.5*a*lambda*Uinflow; % BE step unew(2:Nh) = A\F; unew(1) = Uinflow; end • Upwind unew(2:Nh) = (1-CFL)*u(2:Nh) + CFL*u(1:Nh-1); % Boundary condition unew(1) = Uinflow; % Inflow • Lax-Friedrich unew(2:Nh-1) = 1/2*(u(3:Nh)+u(1:Nh-2)) - CFL/2*(u(3:Nh) - u(1:Nh-2)) % Boundary condition unew(1) = Uinflow; % Inflow unew(Nh) = (1-CFL)*u(Nh) + CFL*u(Nh-1); % Outflow • Lax-Wendroff k = a^2 * dt^2 unew(2:(Nh-1)) = u(2:Nh-1) - CFL/2 * ( u(3:Nh) - u(1:Nh-2) ) ... + k/(2*h^2) * ( u(3:Nh) - 2*u(2:Nh-1) + u(1:Nh-2) ) % Boundary condition unew(1) = Uinflow; % Inflow unew(Nh) = (1-CFL)*u(Nh) + CFL*u(Nh-1); % Outflow N.B. In LW scheme: Upwind: k = abs(a) * h * dt Law-Friedrich: k = h^2 Heat eq.: A = (h^(-2))*spdiags([-e, 2*e, -e], [-1, 0, 1], M-1, M-1); % Initial and boundary conditions u(:, 1) = y0(x, t0); u(1, :) = 0; u(end, :) = 0; % Theta method applied to the internal nodes I = speye(M-1); for n = 1:N u(2:end-1, n+1) = ... (I + dt*theta*A) \ ((I - dt*(1-theta)*A)*u(2:end-1, n)); end • FE theta = 0 • BE theta = 1 • CN theta = 1/2 Poisson eq.: • Dirichlet BC A = spdiags([-e, 2*e, -e], [-1, 0, 1] , M-1, M-1)/(h^2); F = f(x(2:end-1))';…