Differential Equations Lecture Work Solutions 333

Differential Equations Lecture Work Solutions 333 - 2....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 2. Apply the Beam-Warming scheme with Euler implicit time differencing to the linearized Burgers’ equation on the computational grid given in Figure 61 (use c = 2, µ = 2, ∆x = 1) and determine the steady state values of u at j = 2 and j = 3. the boundary conditions are u n = 1, 1 u 1 = 0, 2 and the initial conditions are un = 4 4 u1 = 0 3 Do not use a computer to solve this problem. un+1 − un j j + cux ∆t Let ν = c n+1 j n+1 = µuxx j ∆t ∆t and r = µ 2 then ∆x ∆x ν n+1 +1 +1 +1 u − un−1 + r un−1 − 2un+1 + un+1 un+1 = un − j j j j j j 2 j +1 or − ν ν +1 +1 + r un−1 + (2r + 1)un+1 + − r un+1 = un j j j j 2 2 In our case c = 2, µ = 2, ∆x = 1 and so if we let ∆t = 1, then r = ν = 2. Using these values in the above equation, we get +1 +1 −3un−1 + 5un+1 − un+1 = un j j j j For n = 1 (don’t forget to employ the boundary conditions) −3 · 1 + 5u2 − u2 = 0 2 3 −3u2 + 5u2 − 4 2 3 =0 The solution of this system of two equations is u2 = 2 19 22 u2 = 3 29 22 Now go to the next time step n = 2 = 19 22 −3u3 + 5u3 − 4 = 2 3 29 22 −3 + 5u3 − u3 2 3 The solution of this system of two equations is 333 ...
View Full Document

This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

Ask a homework question - tutors are online