Numerics for PDEs

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FB 3 | Mathematical Institute
Numerics for PDEs
Prof. Dr. Thomas G¨otz
Moritz Sch¨afer
Tutorial 2
February 25, 2021
Exercise 8:
tive.
(Theoretical) Show, that the upwind method is first order consistent and conserva
Exercise 9: (Theoretical) Show, that the shock solution
u(t; x) = (1 : 0 : xx > t= ≤ t=22
is a weak solution of the Burgers equation ut + uux = 0 with initial condition
u(0; x) = (1 : 0 : xx >≤ 0 0 :
Exercise 10: (Numerical Lab) Implement a python function total_variation(u, h) computing
the total variation of a grid function u on an equidistant grid of step size h.
Exercise 11: (Numerical Lab) Apply the upwind method to the Burgers equation ut + uux = 0
on the spatial interval x 2 [-5; 5] with initial condition
(a) (Shock solution):
u(0; x) = (1 : 0 : xx >≤ 0 0 :
(b) (Rarefaction wave):
u(0; x) = (0 : 1 : xx >≤ 0 0 :
Check your numerical results for different space steps h and different grid ratios γ = τ=h.
Also check the total variation of the solution using the function implemented in the previous
exercise.
To be discussed in week 17 1/1

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