MATLAB Mathematics - SERC - Index of

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5 Differential Equations 2 u′ i = 1 + u i vi – 4u i + α( N + 1) 2 ( u i – 1 – 2u i + u i + 1 ) 2 v′ i = 3u i – u i vi + α( N + 1) 2 ( v i – 1 – 2v i + v i + 1 ) and is solved on the time interval [0,10] with α = 1/50 and u i ( 0) = 1 + sin( 2πx i ) v i ( 0) = 3 ⎭ ⎬⎫ with x i = i ⁄ ( N + 1) , for i = 1, ..., N There are 2N equations in the system, but the Jacobian is banded with a constant width 5 if the equations are ordered as u 1 , v 1 , u 2 , v 2 , … In the call brussode(N), where N corresponds to N , the parameter N ≥ 2 specifies the number of grid points. The resulting system consists of 2N equations. By default, N is 20. The problem becomes increasingly stiff and the Jacobian increasingly sparse as N increases. The nested function f(t,y) returns the derivatives vector for the Brusselator problem. The subfunction jpattern(N) returns a sparse matrix of 1s and 0s showing the locations of nonzeros in the Jacobian ∂f ⁄ ∂y . The example assigns this matrix to the property JPattern, and the solver uses the sparsity pattern to generate the Jacobian numerically as a sparse matrix. Providing a sparsity pattern can significantly reduce the number of function evaluations required to generate the Jacobian and can accelerate integration. For the Brusselator problem, if the sparsity pattern is not supplied, 2N evaluations of the function are needed to compute the 2N-by-2N Jacobian matrix. If the sparsity pattern is supplied, only four evaluations are needed, regardless of the value of N. To run this example, click on the example name, or type brussode at the command line. From the command line, you can specify a value of N as an argument to brussode. The default is N = 20. function brussode(N) %BRUSSODE Stiff problem modeling a chemical reaction if nargin < 1 N = 20; end 5-26
Initial Value Problems for ODEs and DAEs tspan = [0; 10]; y0 = [1+sin((2*pi/(N+1))*(1:N)); repmat(3,1,N)]; options = odeset('Vectorized','on','JPattern',jpattern(N)); [t,y] = ode15s(@f,tspan,y0,options); u = y(:,1:2:end); x = (1:N)/(N+1); surf(x,t,u); view(-40,30); xlabel('space'); ylabel('time'); zlabel('solution u'); title(['The Brusselator for N = ' num2str(N)]); % -------------------------------------------------------------- function dydt = f(t,y) c = 0.02 * (N+1)^2; dydt = zeros(2*N,size(y,2)); % preallocate dy/dt % Evaluate the two components of the function at one edge of % the grid (with edge conditions). i = 1; dydt(i,:) = 1 + y(i+1,:).*y(i,:).^2 - 4*y(i,:) + ... c*(1-2*y(i,:)+y(i+2,:)); dydt(i+1,:) = 3*y(i,:) - y(i+1,:).*y(i,:).^2 + ... c*(3-2*y(i+1,:)+y(i+3,:)); % Evaluate the two components of the function at all interior % grid points. i = 3:2:2*N-3; dydt(i,:) = 1 + y(i+1,:).*y(i,:).^2 - 4*y(i,:) + ... c*(y(i-2,:)-2*y(i,:)+y(i+2,:)); dydt(i+1,:) = 3*y(i,:) - y(i+1,:).*y(i,:).^2 + ... c*(y(i-1,:)-2*y(i+1,:)+y(i+3,:)); % Evaluate the two components of the function at the other edge % of the grid (with edge conditions). i = 2*N-1; dydt(i,:) = 1 + y(i+1,:).*y(i,:).^2 - 4*y(i,:) + ... c*(y(i-2,:)-2*y(i,:)+1); 5-27
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6 Sparse Matrices Function Summary
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Index LU factorization 6-30 minimum
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