High order well balaced schemes for systems of - Philippe LeFloch

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Practical considerationsIntegrals on each interval appearing in the non-linearEquations for the reconstructions, e.g. for WENO 2-3are replaced by Gaussian quadrature formulas.[We used 4 point Gauss-Legendre formulas][which in practice guarantee WB property within round offError in our tests]WB error depends on the tolerance for the solution of thenonlinear equationsOther reconstruction techniques are possible (see last part)
Unstaggered gridsskipEvolution equationNumerical flux functionBoarder valuesWith (WENO 2-3)Source average
Page 1 and 2: High order well balanced schemesfor
Page 3 and 4: Outline• Well balanced schemes•
Page 6 and 7: Some references (not complete!)Bern
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 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

High order well balaced schemes for systems of - Philippe LeFloch

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