High order well balaced schemes for systems of - Philippe LeFloch

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Some references (not complete!)Bernudez, A., Vazquez, M.E., 1994, Computer Fluids.L. Gosse, LeRoux, 1996, WB scheme for scalarGreenberg, LeRoux, 1996, … non conservative productsR. Le Veque, 1998, WB Godunov scheme based on wave propagationG.R., HYP2000, WB central scheme (staggered)Perthame and Simeoni, 2001, WB kinetic schemeS.Jin, 2001, WB FV scheme for systems with geometric sourceKurganov and Levy, 2002, WB central unstaggeredG.R., 2002, central staggered, preserve non static equilibriaBouchut, 2004, nonlinear stability of finite volume, Birkhauser...Noelle, Pankratz, Puppo, Natvig, 2007, High order, FV well balancedNoelle, Shu, Xing, 2007, high order WB WENO FV schemeGallardo, Pares, Castro, JCP, 2007, high order, wb sw topography & dry areasKhe and G.R., 2008, Hyp. conference, Maryland
A second order WB scheme for shallow waterPrototype of hyperbolic system with source term.Probably the most studied case.Valid when the wavelength >> water depth.Pressure is hydrostatic, and horizontal velocity does notdepend on vertical coordinate.Equations:
Page 1 and 2: High order well balanced schemesfor
Page 3 and 4: Outline• Well balanced schemes•
Page 9 and 10: When and why does it work?• The m
Page 11: Conservative and equilibrium variab
Page 14 and 15: Define equilibriumcell averageFirst
Page 16 and 17: High order in space:conservative re
Page 18 and 19: Computation of the stage values 1/2
Page 20 and 21: Practical considerationsIntegrals o
Page 22 and 23: Forward EulerLet us show WB propert
Page 24 and 25: Numerical tests:staggered FV scheme
Page 26 and 27: Shallow water - static equilibiumN
Page 28 and 29: Shallow water - moving equilibiumN
Page 30 and 31: Equilibrium preservationShallow wat
Page 32 and 33: Non‐staggered schemes for shallow
Page 34 and 35: 1 st order scheme• Left: h, Right
Page 36 and 37: Accuracy Tests• Tests performed:
Page 38 and 39: Conservative and Equilibrium variab
Page 40 and 41: Equilibrium preservationNozzle flow
Page 42 and 43: Application to ADER schemesThe proc
Page 44 and 45: Using the propertyADER WB schemesth
Page 46 and 47: Equilibrium statesDefined by (1)whe
Page 48 and 49: First order schemeStaggered schemeI
Page 50 and 51: They satisfy the collocationequatio
Page 52 and 53: Time advancement (staggered version
Page 54 and 55: Second order scheme
Page 56 and 57:
Test with a smooth solutionCheck fo
Page 59 and 60:
Shallow water system4 th Order Sche
Page 61 and 62:
Total heightReconstruction: restDis
Page 63 and 64:
Numerical Tests. OrderL1-norm of er
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High order well balaced schemes for systems of - Philippe LeFloch

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