Numerical Methods for Partial Differential Equations . CAAM 452 Spring 2005 Lecture 7 Instructor: Tim Warburton. Summary of Small Theta Analysis. The dominant remainder term in this analysis relates to a commonly used, physically motivated description of the shortfall of the method:.
Instructor: Tim Warburton
After 4 periods the solution is totally flattened in all but the last 2 results. If we botheredto keep increasing M we would eventually see the error decline as 1/M
We clearly see that there is initial growth
near the pulses, but eventually the
dominant feature is the highly oscillatory
and explosive growth
(large m in the above red term).
What should we use as an error
Q1) Build a finite-difference solver for
Q1a) use your Cash-Karp Runge-Kutta time integrator from HW2 for time stepping
Q1b) use the 4th order central difference in space (periodic domain)
Q1c) perform a stability analysis for the time-stepping (based on visual inspection of the CK R-K stability region containing the imaginary axis)
Q1d) bound the spectral radius of the spatial operator
Q1e) choose a dt well in the stability region
Q1f) perform four runs with initial condition
(use M=20,40,80,160) and compute maximum error at t=8
Q1g) estimate the accuracy order of the solution.
Q1h) extra credit: perform adaptive time-stepping to keep the local truncation error from time stepping bounded by a tolerance.