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The Finite Volume Method

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  1. The Finite Volume Method FluxLimitersTVD Ingo Philipp ComputationalAstrophysics

  2. Integral Form impermeable wall x1 x2 flowtotheright flowtotheleft substanceneither creatednordestroyedin [x1, x2] mass in [x1, x2] at time t2 > t1 in termsofthe total massat time t1 & the total (integrated) fluxateachboundaryduring [t1, t2] integral form oftheconservationlaw!

  3. Differential Form differential form r(x,t) and v(x,t) aredifferentiablefunctions, i.e. thisdoesn‘t hold ifthedensity isdiscontinuous The integral form ismore fundamental physically andthustheappropriaterepresentation integral form continuestobe valid evenfordiscontinuoussolutions

  4. General Form a dS outflowdefines a lost ofsomesubstance! outflow inflow integral form differential form: balancelaw differential form

  5. The Finite Volume Method Ui Ui+1 Ui-1 Vi stationarymesh – constantDx integral lawistransferredto smallcontrolvolumes vertexcentered xi-1 xi xi+1 xi-1/2 xi+1/2 with piecewiseconstantcellaverageforeach U trueforany integrateoversmall time stepDt – howdoesthecellmeanevolve in time? x1 xN meanevolutionequationwithoutapproximation

  6. The Finite Volume Method (0,1) yj+1/2 Dy (1,0) (-1,0) (i,j) Dx yj-1/2 xi+1/2 xi-1/2 1D & withoutsourcesQ 2D & fluxapproximation quadrature – mid-pointrulewith thetruefluxattheinterfacesisreplacedby a numericalfluxfunctionbased on (0,-1)

  7. PiecewiseLinear Reconstruction use the cell averages to compute a polynomial representation of U for each cell the easy way out: polynomial of 0th order we could instead assume a linear behavior for for where xi+1 xi-1 xi the average value of over the control volume is regardless of the slope

  8. The Advection Equation solve with IC and newsetof variables andgiveswith profiledoesn‘tchange in shape – itshifts in positivev>0or in negativev<0direction t characteristic tn x characteristic x t DODforU(x,t)is just the singlepoint(x-v(t-tn), tn) t x

  9. The Advection Equation sinceweknowtheanalyticalsolutionweareabletocomputethefluxintegrals (numericalfluxfunctions) withthehelpofthepolynomialreconstructed , i.e. and characteristics outflow – backtrackintoithcellatthenthtime level inflow – backtrackinto (i-1)thcellatthenthtime level tn+1 for characteristics tn xi-1 xi xi+1

  10. The Advection Equation tn+1 tn xi-1 xi xi+1

  11. Choice ofSlopes upwind (Godunov‘smethod) centeredslope (Fromm) upwindslope (Beam-Warming) downwindslope (Lax-Wendroff) numerical DOD containsphysical DOD & von Neumann stable if

  12. Upwind artificialdiffusion upwind (Godunov‘smethod) U(x,t) x numericaldiffusion localdiscretizationerror

  13. Lax Wendroff artificialdiffusion downwindslope (Lax-Wendroff) U(x,t) x numericaldispersion localdiscretizationerror

  14. Beam Warming upwindslope (Beam-Warming) U(x,t) x numericaldispersion localdiscretizationerror

  15. Beam Warming upwindslope (Beam-Warming) periodicboundarycondition U(x,t) x

  16. Fromm centeredslope (Fromm) U(x,t) x numericaldissipation localdiscretizationerror

  17. Whatwent wrong ? downwindslope (Lax-Wendroff) appliedto 1 initialprofile 0 J J+1 J-1

  18. Whatwent wrong ? downwindslope (Lax-Wendroff) appliedto 1 0 J J+1 J-1

  19. Whatwent wrong ? downwindslope (Lax-Wendroff) appliedto 1 0 J J+1 J-1 tn+2 tn+1 tn J J+1 J J+1 J-1

  20. Whatwent wrong ? downwindslope (Lax-Wendroff) appliedto overshoot 1.125 1 overshoot 0.375 0 J J+1 J-1 tn+2 tn+1 tn J J+1 J J+1 J-1

  21. Whatwent wrong ? downwindslope (Lax-Wendroff) appliedto 1.172 1 0.98 0.7 0.14 initialprofile 0 J J+1 J-1 overshoot overshoot overshoot undershoot

  22. Whatwent wrong ? any negative slope in theJthcellleadsto a volumeaverage > 1attn+1 toavoidoscillations just settheslopetozero gives1st order upwindmethod but in smooth regionswewant2ndorder accuracy (Lax-Wendroff) benefitfromboth • near a discontinuitywemaywanttolimittheslope • in smooth regionswechoosesth. liketheLax-Wendroffslope …howmuchshouldwelimittheslope? …howtocontroltheflux? …how do wemeasureoscillations in thesolution? TOTAL VARIATION

  23. Flux Limiter …howtocontroltheflux? the time averagedfluxattheinterface should now be determined by the jump xi-2 xi+1 xi xi-1 givesus a jump in smooth regionsand limited versionofthe jump farfrom 1 near a discontinuity wemightwant a fluxlimiterffunctionthat hasvaluesnear 1 forq~1, but thatreduces orincreasestheslopewherethedatais not smooth fluxlimiterfunction for for measureofthesmoothnessofthedatanear

  24. Flux Limiter upwind (Godunov‘smethod) centeredslope (Fromm) upwindslope (Beam-Warming) downwindslope (Lax-Wendroff)

  25. *High Resolution Schemes for Hyperbolic Conservation Laws Total Variation Howdoesf(x)vary on [a,b]? supremumof sums over all partitions toavoidoscillationswerequirethatthemethod doesn‘tincreasethe total variation(TVNI) 1 foranystartingdata Amiram Harten*(1983) 0 p 2p a monotone scheme is TVNI if initial condition is then a TVNI scheme is monotonicity preserving -1 Godunov‘stheorem monotone schemes can be at most 1st order accurate

  26. Harten’s Theorem may in generalbedatadependent THEOREMForanyschemeoftheabove form, a sufficientconditionforthescheme tobe TVNI isthatthecoefficientssatisfy advectionequation CFL for all valuesofand and ifweareat an extremumandweshouldtake Osher and Chakravarthy(1984) TVD schemes must degenerate to 1st order accuracy at extremal points

  27. * High resolution schemes using flux-limiters for hyperbolic conservation laws Total Variation and P.K. Sweby*(1984) Fromm Fromm Beam-Warming Beam-Warming 2 2 Lax-Wendroff Lax-Wendroff 1 1 TVNI 2nd order TVNI Godunov 2 Godunov 2 1 3 1 3 noneoftheselinearlimitersgenerate a TVNI scheme any2nd order scheme relying on must be a weighted average of the LW and BW scheme

  28. MinMod 2 U(x,t) minmod 1 2nd order TVNI 2 1 3 x slopelimiterversion Lax-Wendroff downwindslope Beam-Warming upwindslope Godunov‘smethod upwind

  29. Monotonized Central Difference 2 MC U(x,t) 1 2nd order TVNI 2 1 3 x slopelimiterversion ~Lax-Wendroff downwindslope Godunov‘smethod upwind Fromm centeredslope ~Beam-Warming upwindslope

  30. References

  31. Upwind&CFL tn+1 tn+1 updatingscheme=upwind informationtravelsmorethanonegridcell in one time step necessary CFL stabilitycondition fulfilled tn tn informationtravelslessthanonegridcell in one time step upwindmethodcertainlyunstable!

  32. Numerical Solution centeredslope (Fromm) upwind (Godunov‘smethod) U(x,t) upwindslope (Beam-Warming) downwindslope (Lax-Wendroff) x

  33. Lax Wendroff downwindslope (Lax-Wendroff) periodicboundarycondition U(x,t) x