UNIVERSITY
Department DREXEL of Civil & Architectural Engineering Computational Hydraulics I CIVE 665 Homework 3 Assigned: 02/06/2007
Problem 1) Consider the first-order wave equation u u +c =0 t x
Due: 02/13/2007
c0
As shown in class, a discretization based on the LAX-method (an average value for uin = 1 (uin+1 + uin-1 ) in the time derivative) leads to a conditionally stable scheme. Show that 2 using a forward step in time and the usual 2nd -order centered difference approximation for the spatial derivative, leads to an unconditionally unstable scheme.
Problem 2) Again, consider the first order wave equation, and the same first order explicit finite difference approximation (forward in time) as in Problem 1. Instead of using an average value for one of the time derivative terms (at time level n), try an "upwind" difference for the spatial derivative, where u 1 n = ui - uin-1 x x
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a) Derive the modified equation determine and whether the scheme is consistent; also determine the accuracy of the scheme. Rewrite the time derivative of u in the error term as a space derivative (using the original equation) and notice that the modified equation now looks like a diffusion equation with a diffusion coefficient that depends on t. What happens as t becomes large? b) Find the stability of this scheme by using a von Neuman stability analysis.
Problem 3) Consider the following finite difference approximation for the diffusion equation t n uin +1 = uin -1 + 2 ui +1 - uin +...