High order well balaced schemes for systems of - Philippe LeFloch

High order well balanced schemesfor hyperbolic systems with sourceGiovanni RussoDepartment of Mathematicsand Computer ScienceUniversity of Catania, ItalyIn collaboration with Alexander KheLavrentyev Institute for hydrodynamicsNovosibirsk, Russia

General features of the new approachGoal: introduce a methodology to derive• Well balanced property for static and moving equilibria• High order accuracy (depends on WENO)• Applicable to a wide class of systems with source• Applicable to staggered and unstaggered finite volume schemes• At a numerical level requires the solution of local equilibria• Conceptually simple

Outline• Well balanced schemes• Conservative and equilibrium variables• Staggered finite volume• Non staggered FV schemes• Conservative reconstruction of equilibrium variables• Numerical tests:– Shallow water equations– Nozzle flow• Well balanced ADER schemes• Numerical reconstruction of equilibrium states• Work in progress

Well balanced schemesConsider a system of balance laws:Denote by the stationary solution, satisfying equil. Eqn.A method is well balanced if it satisfies a discrete versionof the equilibrium equation.If a method is not well balanced, truncation error of solutionsenear equilibrium may be larger than u( x,t)− u ( x)

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:

When and why does it work?• The method preserves static equilibriabecause if w = 0 then H = h + B is constant.• The method works because the unknownvariables are also equilibrium variables.• It would be nice to use equilibrium variablesfor the evolution, but then one would looseconservation.• Key point: conservative mapping betweenequilibrium and conservative variables.

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)

Forward Euler in timeFirst order schemeHow to do it in such a way that the scheme is conservativeand well balanced?Basic idea: use conservative variable for the evolution,and equilibrium variables to help the reconstruction

Define equilibriumcell averageFirst order scheme - 2Note: This definition has been used by Noelle, Shu and XingThen define the needed quantities as:Staggered cell valuesPointwise valuesStaggered source averages

Well balanced propertyIt is easy to show that, if { un j} are cell averages of equilibriumsolution, thenandLet be an equilibrium solution, and let be thecorresponding equilibrium variable. ThenIntegrating the equilibrium equation over the space:Making use of this relation in the evolution for the cell average:

High order in space:conservative reconstructionConsider WENO 2-3 in cell jConservative variable u is reconstructed asObtained by imposing correctcell averages in cells j and j + 1 :Set of nonlinear equations for andSimilarly forRemark: conservation property of the mapping is neededto ensure that v is actually constant if the set { u j} comesfrom equilibrium solution, not to ensure that the scheme isconservative!

Time integration: central Runge-KuttaEvolution equationCRK approach: numerical solution on the staggered cellStage valuescomputed by theevolution equation in non conservative form on the edgeof the staggered cell (i.e. at the center of the cells)

Computation of the stage values 1/2Observe thatwhereA= ∇ufConsider the evolution equationAt equilibriumThis relation in fact can be used to define theequilibrium variable vEvolution equation for the stage values:

Computation of the stage values 2/2Runge-Kutta coefficientsThe term can be obtained using WENOon the derivatives of the reconstructionThe WENO reconstruction is obtained frompointwise reconstruction of v obtained from the stage value

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

Forward EulerLet us show WB propertyAssume equilibrium initial dataThenTherefore, by consistency:Source cell averageAbove relations implyAnd therefore

High order schemesCan be obtained by applying RK schemes in timeNumerical solutionStage valuesValues at cell edges:[conservative reconstruction]Using same argument as Euler scheme, it can be proved it is WB,i.e. because it is proportional to derivatives of

Numerical tests:staggered FV schemesSchemes compared• First order: piecewise constant,forward Euler• Second order: piecewise linear, MM limiter,CRK2 (modified Euler)• Third order: compact WENO,CRK3• Fourth order:Central WENO (3 parabolas),CRK4

Numerical tests• Models considered:– Saint Venant equations of shallow water– Nozzle flow for Euler equations• Test performed to check– WB property for• Static equilibria• Moving equilibria– Order of accuracy– Shock capturing property

Shallow water – static equilibiumN = 200, 3200First orderSecond orderh ( x , t 1)q ( x , t 1)h ( x , t 1)q x , t )(1

Shallow water – static equilibiumN = 200,3200Fourth orderThird order

Shallow water – moving equilibiumN = 200, 3200First orderSecond orderh ( x , t 1)q ( x , t 1)h ( x , t 1)q x , t )(1

Shallow water – moving equilibiumN = 200, 3200Third orderFourth orderh ( x , t 1)q ( x , t 1)h ( x , t 1)q x , t )(1

Equilibrium preservationShallow water, Moving equilibrium, 1st order schemeL1 errors in h and q, dt/dx = 0.2Cells: 100: 2.7693E-010 1.9099E-010Cells: 200: 5.4388E-010 3.8929E-010Cells: 400: 1.0930E-009 7.8746E-010Shallow water, Moving equilibrium, 4th order schemeL1 errors in h and q, dt/dx = 0.2Cells: 100: 2.6651E-013 4.2220E-014Cells: 200: 5.6666E-013 3.8355E-014Cells: 400: 1.4794E-011 6.9670E-012

Accuracy test (smooth solution)Skip non staggered

Non‐staggered schemes for shallow waterNumerical Flux —HLL Riemann SolverCFL number = 0.91 st order scheme2 nd order scheme:o Piecewise linear reconstruction with MinMod limitero Modified Euler4 th order scheme:o WENO parabolic reconstructiono Runge—Kutta 4Number of cells: 100 and 3200

Moving Equilibrium1.2• Initial ConditionsFinal time = 0.3810.80.6hx ( )⎧H( x, q0, ε0) + 0.001, x∈[0.45,0.55]= ⎨⎩ H( x, q0, ε0), otherwise0 00.40.2u = q / h ( x), x∈[0,1]00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1bx ( )( π x )⎧⎪ 0.25 1+ cos 20 ( −0.65) , x∈[0.6,0.7]= ⎨ ⎪ ⎩0.0, otherwise

1 st order scheme• Left: h, Right: q = rho u•Solid: 3200 cells, Crosses: 100 cells1.0011.00081.0006Initial3200 cells100 cells0.20050.20040.20030.20020.20011.00041.00020.20.19990.199810.99980 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.19970.1996Initial3200 cells100 cells0.19950 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

4 th order scheme• Left: h, Right: q = rho u•Solid: 3200 cells, Crosses: 100 cells1.0011.00081.0006Initial3200 cells100 cells0.20050.20040.20030.20020.20011.00041.00020.20.19990.199810.99980 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.19970.1996Initial3200 cells100 cells0.19950 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Accuracy Tests• Tests performed:• 100, 200, 400, 800, 1600, 3200 cellsh q0.879 0.9800.934 0.9840.970 0.9950.984 0.997h q1.837 1.8861.913 1.8861.899 1.9551.764 1.797h q3.980 3.9444.060 4.0464.136 4.1234.094 4.087

Nozzle flowEuler equations on a channel of variable cross sectionCross section of the channelInitial conditionPeriodic B.C.CFL = 0.2Final time T=0.26

Conservative and Equilibrium variablesConservativeEquilibrium~A new equilibrium variable S = S / S 0( x)is used in place of S.~The new variable is S =1 at equilibrium.Calculations are performed with staggered grid

Nozzle flow – shock capturing and WBPressureVelocitySecond orderFirst order

Equilibrium preservationNozzle flow, Moving equilibrium, 1st order schemeL1 errors in conservative variables, dt/dx = 0.2Cells: 100: 2.2272E-016 1.9529E-015 6.7946E-016Cells: 200: 9.9143E-016 3.5666E-015 4.7939E-015Cells: 400: 3.4360E-015 1.6227E-014 3.1442E-015Nozzle flow, Moving equilibrium, 2nd order schemeL1 errors in conservative variables, dt/dx = 0.2Cells: 100: 3.5016E-016 3.7648E-015 7.9936E-016Cells: 200: 7.6102E-016 4.7445E-015 3.1175E-015Cells: 400: 1.5455E-015 2.7246E-014 1.4070E-014

Nozzle flow: accuracy test

Application to ADER schemesThe procedure can be used to construct WB ADER schemes.ADER is a technique introduced by Toro and developed by Toro,Titarev, Dumbser, etc. (recall talk of Castedo Ruiz on monday).In classical ADER schemes, the numerical solution of asystem of the formis obtained as follows: integrating over a cell one obtainsSkip ADER

summary of ADER schemes• The solution at cell edges may be computed by Taylorexpansion in time at cell edges.• Taylor expansion contains time derivative of the solutionevaluated at the initial time.• Differentiating the original equation in space, time derivativescan be computed from space derivatives (Cauchy-Kovalewsky procedure).• The initial value at the cell edge is evaluated by the solutionof the Riemann problem (or by an approximate Riemannsolver)• The initial value of the space derivatives at cell edges isevaluated by the solution of a linear Riemann problem.Remark: Taylor expansion can be replaced by Runge-Kuttaprocedure (G.R., Titarev, Toro, 2006)

Using the propertyADER WB schemesthe evolution equation on cell edges becomes:Now, if are cell averages of an equilibrium solutionthen one hasWhich impliesAnd therefore

Numerical reconstruction of(A.Khe, G.R., in preparation)equilibrium statesThe approach used so far requires the explicit knowledgeof the mapping between conservative and equilibriumvariables.Such a mapping is not always available or known.It would be desirable to formulate the method withoutrelying on the explicit knowledge of the mapping.This can be obtained by a numerical reconstruction ofequilibrium states

Equilibrium statesDefined by (1)whereLocal equilibrium in cell is defined by (1)and(2)A state which is formed by piacewise local equilibrium statesis the natural generalization ofa piecewise constant state forsystems without source( : char. function of , solution of (1) and (2) )

Numerical local equilibriaSolve Eq.(1) and (2) numerically, e.g. by collocation.Look for an approximate solution in a finitedimensional space, say polynomial of degreewhich gives p+1 equations for p+1 unknownsRemarks• By piacewise local equilibria one can construct WBschemes of order one.• The quality of WB property depends on the accuracy ofthe solution of local equilibria (at equilibrium there arejumps )

First order schemeStaggered schemeIt isbecause =0(a part from )Unstaggered schemewhich at equilibrium gives

High order schemesAre obtained by high order reconstructions from local equilibria(rather than from constant states)Second (and third) order recons. (equivalent to WENO2-3)In cell define function aswhereare the WENO weightand the local equilibria , are defined in :

They satisfy the collocationequation in p points inand the cell averageconditionsLinear polynomialsare obtained by imposing thatthe reconstruction is conservative:4x4 linear system

RemarkThis property is satisfied by the first order polynomialsthat matches cell averages on two adjacent cells inclassical FV schemes:

Time advancement (staggered version)Staggered cell average computed from the reconstructionInitial values at cell center computed by from the reconstructionnumerical solution does not change (to )if stage values do not change.Stage values: computed fromRHS = 0 at equilibriumHigher order schemesCan be constructed with a similar procedureHere we construct up to a fourth order method

Application to scalar equationBottom profile: hump centered in 0.6

Second order scheme

Fourth order scheme

Test with a smooth solutionCheck for convergence rate ( error)Second order schemeFourth order scheme

Application to shallow water

Shallow water system4 th Order SchemeI. Lake at rest1.210.80.60.40.201.0110 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Initial configuration1.0051.004Total height0.0060.004Discharge1.0030.0021.00201.001-0.00210.9990 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.004-0.0060 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Solid — 4000 cells, Circles — 100 cellsOne can notice small perturbations over [0.6, 0.7]due to numerical reconstruction

Shallow water system4 th Order SchemeII. Moving Equilibrium1.210.80.60.40.201.0110 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Initial configuration1.0061.0051.0041.0031.0021.00110.999Total height0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2050.2040.203 Discharge0.2020.2010.20.1990.1980.1970.1960.1950 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Solid — 4000 cells, Circles — 100 cellsOne can notice small perturbations over [0.6, 0.7]due to numerical reconstruction

Total heightReconstruction: restDischarge1.0051.0041.0034000 cells400 cells200 cells100 cells0.0060.0040.0021.00201.00110.9990 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.002-0.004-0.0064000 cells400 cells200 cells100 cells0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.00040.00041.00020.0002100.99980.9996400 cells200 cells100 cells-0.0002-0.0004400 cells200 cells100 cells0.55 0.6 0.65 0.7 0.750.55 0.6 0.65 0.7 0.75

Reconstruction: movingTotal heightDischarge1.0061.0051.0041.0031.0021.00114000 cells400 cells200 cells100 cells0.9990 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2050.2040.2030.2020.2010.20.1990.1980.1970.1960.1954000 cells400 cells200 cells100 cells0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.00041.0002400 cells200 cells100 cells0.20150.2010.2005400 cells200 cells100 cells10.20.99980.19950.99960.55 0.6 0.65 0.7 0.750.1990.19850.55 0.6 0.65 0.7 0.75

Numerical Tests. OrderL1-norm of errorOrderCells h q h q64128 1.42e-5 1.63e-5256 5.90e-7 5.86e-7 4.59 4.80512 2.55e-8 2.32e-8 4.53 4.661024 1.19e-9 9.83e-10 4.42 4.562048 6.17e-11 4.80e-11 4.27 4.364096 3.58e-12 2.84e-12 4.11 4.088192 2.20e-13 1.76e-13 4.03 4.01

ConclusionHigh order WB can be obtained using:conservative variables for the evolutionequilibrium variables used in the reconstructionanalytical knowledge of equilibrium variables is notnecessaryWork in progress• Include other effects (dry zones, transcritical flow, etc.)• Apply to other models (stratified athmosphrere, …)• Apply numerical reconstruction to problems with morespace dimensions• Connections with other approaches