Linear Differential Equations with Variable Coefficients by I Z Shtokalo
By I Z Shtokalo
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Additional info for Linear Differential Equations with Variable Coefficients
Example text
For is = l:delr(l) ist = is; 7. 7o Build the perturbation vector. 7. pt = zeros(n,l); while ist <= n pt(ist) = 1; ist = delr(ist)+l; end 7. 7» Compute the forward difference. 7. xl = x+epsnew*pt; fl = feval(f,xl); dv = (fl-fO)/epsnew; ist = is; 7. % Fill the appropriate columns of the Jacobian. 7. 4. The Chord and Shamanskii Methods 33 The internal MATLAB code numjac is a more general finite difference Jacobian code, numjac was designed to work with the stiff ordinary differential equation integrators [68] in MATLAB.
The components of parms are maxit is the upper limit on the nonlinear iteration; the default is 40, which is usually enough. The Jacobian is computed and factored after every isham nonlinear iterations or whenever the ratio of successive norms of the nonlinear residual is larger than rsham. So, for example, isham — 1 and rsham = 0 is Newton's method. 5, so the Jacobian is updated only if the decrease in the nonlinear residual is not sufficiently rapid. 1) of using an out-of-date Jacobian when far from a solution is reduced.
L; nx=63; nt=l+l/dt; dx=l/(nx+l); tval=0:dt:1; xval=0:dx:1; 7. 7. Use tight tolerances, Newton's method, and a tridiagonal Jacobian. y. d-6]; parms=[40, 1, 0, 1, 1, 1]; uhist=zeros(nx+2,nt); uold=zeros(nx,1); for it=l:nt-l [unew, it_hist, ierr] =nsold (uold,' f time', tol, parms) ; uhist(2:nx+1,it+1)=unew; uold=unew; end 7, 7. Plot the results. 7. 4. 13). 50 Chapter 2. Finding the Newton Step with Gaussian Elimination You can see from the plot that u(x, t) tends to a limit as t —> oo. 14) would be to solve the time-dependent problem and look for convergence of u as t —> oo.