High order well balaced schemes for systems of - Philippe LeFloch

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Conservative and equilibrium variables 1/2In many cases, equilibrium (for smooth solutions)some functions of the conservative variables are constant.For example, for shallow water:At equilibrium q ≡ hw = const, andfrom the second equation the following variable is constant:[Energy density – mathematical entropy]
Conservative and equilibrium variables 2/2Some notation:Let us denote by u the conservative variableand by v the equilibrium variable.We shall assume that there is a 1 to 1 correspondenceBetween u and v:u = U ( v,x)⇔ v = V ( u,x)In the case of shallow water, one has⎛ h⎞⎛ q ⎞2 2u = ⎜ ⎟, v = ⎜ ⎟ η(h , q,x)= q /(2h) + g~ ( h + B(x))⎝q⎠ ⎝η⎠Inversion requires solution of a cubic equation (which weassume we can do. Must be careful for transcritical cases)
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: When and why does it work?• The m
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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