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Unformatted text preview: MPO 662 Problem Set 3 1 Hyperbolic equations . Consider the Lax Fredrichs approximation to the scalar advection equation: u n +1 j- u n j +1 + u n j- 1 2 t + c u n j +1- u n j- 1 2 x = 0 Determine the truncation error for this scheme Study the consistency of this FD representation What condition to you have to impose on x so that the time errors and spatial errors are of the same order of magnitude? Determine the stability characteristics of this scheme. 2 Parabolic equations 2.1 Properties of O ( t 2 , x 2 ) approximation to heat equation The 1D heat equation, u t = u xx , is a parabolic equation. It is discretized using an implicit scheme of the form: u n +1 j- u n j t = 1 2 bracketleftBigg u n +1 j +1- 2 u n +1 j + u n +1 j- 1 x 2 + u n j +1- 2 u n j + u n j- 1 x 2 bracketrightBigg (1) Study the stability and convergence of this method for periodic boundary conditions. Hint: expand in Taylor series about ( j x, ( n + 1 2 ) t ), do the space-expansion first, and use it in analyzing the time-series. 2.2 stability of O ( t, x 4 ) approximation to heat equation Study the stability restrictions, if any, for the following approximation of the heat equation: u n +1 j- u n j t = - ( u n j +2 + u n j- 2 ) + 16( u n j +1 + u n j- 1 )- 30 u n j 12 x 2 (2) 2.3 stability of hyperviscous operator Study the stability limit of the following approximation to a hyperviscous evolution equation u t =- u xxxx according to u n +1 j- u n j t =- ( u n j +2 + u n j- 2 )- 4( u n j +1 + u n j- 1 ) + 6 u n j x 4 (3) 3 Finite Volume Methods We will write a code to solve numerically the advection-diffusion equation using a finite volume discretization in space. We will do so in stages so as to verify the procedures put in place. The equation is: h t + F x = D x , (4) where F = ( uh ) and D = h x are the advective and diffusive fluxes, respectively. Here u is the advective velocity and...
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- Spring '08