Notes: Burgers' equation Notes - UW Faculty Web Server

1 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.]

2 ]Riemann problem for Burgers equationut+(12u2)x= 0, ut+uux= (u) =12u2,f (u) = Riemann problem with statesu` anyu`, ur, there is a weak solution consisting of thisdiscontinuity propagating at speed given by theRankine-Hugoniot jump condition:s=12u2r 12u2`ur u`=12(u`+ur).Note: Shock speed is average of characteristic speed on might not be the physically correct weak solution! LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Burgers equationThe 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. ]Weak solutions to Burgers equationut+(12u2)x= 0, u`= 1, ur= 2 Characteristic speed:uRankine-Hugoniot speed:12(u`+ur).

3 Physically correct rarefaction wave LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Weak solutions to Burgers equationut+(12u2)x= 0, u`= 1, ur= 2 Characteristic speed:uRankine-Hugoniot speed:12(u`+ur).Entropy violating weak LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Weak solutions to Burgers equationut+(12u2)x= 0, u`= 1, ur= 2 Characteristic speed:uRankine-Hugoniot speed:12(u`+ur).Another Entropy violating weak LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Vanishing viscosity solutionWe wantq(x,t)to be the limit as 0of solution toqt+f(q)x= selects a unique weak solution: Shock iff (ql)> f (qr), Rarefaction iff (ql)< f (qr).

4 Lax Entropy Condition:A discontinuity propagating with speedsin the solution of aconvex scalar conservation law is admissible only iff (q`)> s > f (qr), wheres= (f(qr) f(q`))/(qr q`).Note: This means characteristics must approach shock fromboth sides astadvances, not move away from shock! LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Riemann problem for scalar nonlinear problemqt+f(q)x= 0with dataq(x,0) ={qlifx <0qrifx 0 Piecewise constant with a single jump Burgers or traffic flow with quadratic flux, the Riemannsolution consists of: Shock wave iff (ql)> f (qr), Rarefaction wave iff (ql)< f (qr).Five possible LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.]}

5 ]Transonic rarefactionsSonic point:us= 0for Burgers sincef (0) = Riemann problem datau`= <0< ur= this case wave should spread in both LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Transonic rarefactionsEntropy-violating approximate Riemann solution:s=12(u`+ur) = goes only to right, no update to cell average on LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Transonic rarefactionsIfu`= urthen Rankine-Hugoniot speed is 0:Similar solution will be observed with Godunov s methodif entropy-violating approximate Riemann solver LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.

6 ]Entropy-violating numerical solutionsRiemann problem for Burgers equation att= 1withu`= 1andur= 2: 3 2 10123 1 with no entropy fix 3 2 10123 1 with entropy fix 3 2 10123 1 resolution with no entropy fix 3 2 10123 1 resolution with entropy LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Approximate Riemann solversFor nonlinear problems, computing the exact solution to eachRiemann problem may not be possible, or too the nonlinear problemqt+f(q)x= 0is approximated byqt+Ai 1/2qx= 0, q`=Qi 1, qr=Qifor some choice ofAi 1/2 f (q)based on dataQi 1, linear system for i 1/2:Qi Qi 1= p pi 1/2rpi 1 1/2= pi 1/2rpi 1/2propagate with speedsspi 1/2,rpi 1/2are eigenvectors ofAi 1/2,spi 1/2are eigenvalues ofAi 1 LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.

7 ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Approximate Riemann solversqt+ Ai 1/2qx= 0, q`=Qi 1, qr=QiOften Ai 1/2=f (Qi 1/2)for some choice ofQi 1 general Ai 1/2= A(q`, qr).Roe conditions for consistency and conservation: A(q`,qr) f (q )asq`,qr q , Adiagonalizable with real eigenvalues, For conservation in wave-propagation form, Ai 1/2(Qi Qi 1) =f(Qi) f(Qi 1). LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Approximate Riemann solversFor a scalar problem, we can easily satisfy the Roe condition Ai 1/2(Qi Qi 1) =f(Qi) f(Qi 1).by choosing Ai 1/2=f(Qi) f(Qi 1)Qi Qi 1/2= 1ands1i 1/2= Ai 1/2(scalar!).Note: This is the Rankine-Hugoniot shock shock waves are correct,rarefactions replaced by entropy-violating LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.

8 ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Approximate Riemann solverQn+1i=Qni t x[A+ Qi 1/2+A Qi+1/2].For scalar advectionm= 1, only one 1/2= Qi 1/2=Qi Qi 1andsi 1/2=u,A Qi 1/2=s i 1/2Wi 1/2,A+ Qi 1/2=s+i 1/2Wi 1 scalar nonlinear: Use same formulas withWi 1/2= Qi 1/2andsi 1/2= Fi 1/2/ Qi 1 to modify these by an entropy fix in the trans-sonicrarefaction LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Entropy fixQn+1i=Qni t x[A+ Qi 1/2+A Qi+1/2].Revert to the formulasA Qi 1/2=f(qs) f(Qi 1)left-going fluctuationA+ Qi 1/2=f(Qi) f(qs)right-going fluctuationiff (Qi 1)<0< f (Qi).High-resolution method: still define waveWand speedsbyWi 1/2=Qi Qi 1,si 1/2={(f(Qi) f(Qi 1))/(Qi Qi 1)ifQi 16=Qif (Qi)ifQi 1= LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.}

9 ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Godunov flux for scalar problemThe Godunov flux function for the casef (q)>0isFni 1/2= f(Qi 1)ifQi 1> qsands >0f(Qi)ifQi< qsands <0f(qs)ifQi 1< qs< minQi 1 q Qif(q)ifQi 1 QimaxQi q Qi 1f(q)ifQi Qi 1,Heres=f(Qi) f(Qi 1)Qi Qi 1is the Rankine-Hugoniot shock LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Entropy-violating numerical solutionsRiemann problem for Burgers equation att= 1withu`= 1andur= 2: 3 2 10123 1 with no entropy fix 3 2 10123 1 with entropy fix 3 2 10123 1 resolution with no entropy fix 3 2 10123 1 resolution with entropy LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.

10 ]Entropy (admissibility) conditionsWe generally require additional conditions on a weak solutionto a conservation law, to pick out the unique solution that isphysically gas dynamics: entropy is constant along particle paths forsmooth solutions, entropy can only increase as a particle goesthrough a functions: Function ofqthat behaves like physicalentropy for the conservation law being : Mathematical entropy functions generally chosen todecrease for admissible solutions,increase for entropy-violating LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ] LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec. ]Entropy functionsA scalar-valued function : lRm lRis a convex function ofqif the Hessian matrix (q)with(i,j)element ij(q) = 2 qi qjis positive definite for allq, , satisfiesvT (q)v >0for allq, v case: reduces to (q)> LeVeque, University of WashingtonIPDE 2011, July 1, 2011 [FVMHP Sec.