Transcription of Notes: Burgers' equation Notes - UW Faculty Web Server
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Notes: Burgers' equation Notes - UW Faculty Web Server
Outline Scalar nonlinear conservation laws Shocks and rarefaction waves Entropy conditions Finite volume methods Approximate Riemann solvers Lax-Wendroff TheoremReading: Chapter 11, LeVeque, University of WashingtonIPDE 2011, July 1, 2011 LeVeque, University of WashingtonIPDE 2011, July 1, 2011 Burgers equationQuasi-linear form:ut+uux= 0 The solution is constant on characteristics so each valueadvects at constant speed equal to the LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Burgers equationEqual-area rule:The area under the curve is conserved with time,We must insert a shock so the two areas cut off are LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.]
Riemann problem for Burgers' equation u t + 1 2 u 2 x = 0 ; ut + uu x = 0 : f (u ) = 1 2 u 2; f 0(u ) = u: Consider Riemann problem with states u ` and u r. Forany u `; ur, there is a weak solution consisting of this discontinuity propagating at speed given by the
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