Numerics for PDEs

FB 3 | Mathematical InstituteNumerics for PDEsProf. Dr. Thomas G¨otzMoritz Sch¨aferTutorial 2February 25, 2021


Exercise 8:tive.


(Theoretical) Show, that the upwind method is first order consistent and conserva


Exercise 9: (Theoretical) Show, that the shock solutionu(t; x) = (1 : 0 : xx > t= ≤ t=22is a weak solution of the Burgers equation ut + uux = 0 with initial conditionu(0; x) = (1 : 0 : xx >≤ 0 0 :Exercise 10: (Numerical Lab) Implement a python function total_variation(u, h) computingthe 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 = 0on 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 previousexercise.To be discussed in week 17 1/1

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