Solving Numerical PDEs: Problems, Applications, Exercises by Luca Formaggia Fausto Saleri & Alessandro Veneziani

Author:Luca Formaggia, Fausto Saleri & Alessandro Veneziani
Language: eng
Format: epub
Publisher: Springer Milan, Milano

(discontinuous). However the backward Euler method smooths the solution with the numerical dissipation, with a low order accuracy. The Crank- Nicolson method is more accurate in time (order 2) and less dissipative, so the initial discontinuities produce oscillations that are not dissipated immediately. In this case the initial data belong to L2(I), and not to H 1(I). Femld approximates u0 with

In particular by computing the L2 norm of u0h we have that blows up for h → 0, while · This approximation in V/, does not fulfill (5.26) and the stability estimate of the numerical solution in H1 is meaningless for h → 0. This results in numerical oscillations in the solution (see Fig. 5.7), which are more evident when the discretization step gets smaller (DI. 5.8).

In practice when the initial data are not regular and it is not possible nor convenient to find a stable approximation of u0 in Vh, it is useful to select θ slightly greater than 0.5. This introduces more dissipation dumping the oscillations and maintains an accuracy of almost second order.

Exercise 5.1.4.

Consider the initial-boundary value problem

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