casa day 9 may, 2006

CASA Day

9 May, 2006

9 May 20069 May 2006 Local Defect Correction for Time-Dependent ProblemsLocal Defect Correction for Time-Dependent Problems 22

OutlineOutline

Transport of passive tracersTransport of passive tracers physical problem, mathematical modelphysical problem, mathematical model

Local Defect Correction (LDC)Local Defect Correction (LDC) basic method and its propertiesbasic method and its properties extensions (conservation, multiple levels of extensions (conservation, multiple levels of

refinement)refinement)

Numerical resultsNumerical results


Transport of passive tracerTransport of passive tracer

Passive tracer:Passive tracer: a contaminant that does not a contaminant that does not

influence the dynamics of the flowinfluence the dynamics of the flow

Goal:Goal: influence of the flow on the tracerinfluence of the flow on the tracer

Applications:Applications: dispersion of pollutantsdispersion of pollutants mixing in chemical reactorsmixing in chemical reactors


Mathematical modelMathematical model

= distribution of passive tracer= distribution of passive tracerPePe = Peclet number= Peclet numberv v = given velocity field (or = given velocity field (or

computed solving computed solving Navier-Stokes)Navier-Stokes)

often has a often has a local high activitylocal high activity solve transport equation using Local solve transport equation using Local

Defect Correction (LDC)Defect Correction (LDC)

inPe

1

t2v


Local Defect Correction (LDC)Local Defect Correction (LDC)

LDC: LDC: adaptiveadaptive method for PDEs method for PDEs with highly localized propertieswith highly localized properties

A coarse grid solution and a fine A coarse grid solution and a fine grid solution are grid solution are iterativelyiteratively combinedcombined

Uniform Uniform structuredstructured grids grids

Hh


Integrate on the Integrate on the coarse gridcoarse grid

Provide boundary conditions locallyProvide boundary conditions locally

Integrate on the Integrate on the local fine gridlocal fine grid

Until convergenceUntil convergence Compute a defect at forward timeCompute a defect at forward time Solve a modified Solve a modified coarse gridcoarse grid problem problem Provide new boundary conditions locallyProvide new boundary conditions locally Integrate on the Integrate on the fine gridfine grid with updated with updated

boundary conditionsboundary conditions

One time step with LDCOne time step with LDC

ttntn-1

ttntn-1

ttntn-1

ttntn-1

Δt

δt


LDC iterationLDC iteration

Coarse grid solution at tn

Fine grid solution at tn

Boundary conditions

Defect


The defectThe defect

PDEPDE

Coarse grid discretizationCoarse grid discretization

Fine grid solution is more accurateFine grid solution is more accurate

DefectDefect

CorrectionCorrection

fLut

u

n,H1n,Hn,HH tt fuuLI

dfuuLI n,H1n,Hn,H1

H tt

0d

n,H1n,Hn,hloc

h,HH tt fuuRLId

0d

n,hlocu


ConvergenceConvergence Unconditionally convergent (i.e. for any Unconditionally convergent (i.e. for any ΔΔt and H)t and H) One or two iterations sufficesOne or two iterations suffices Convergence rate is O(Convergence rate is O(ΔΔtt22 H H-4-4))

with implicit Euler + centered diff.with implicit Euler + centered diff.

Limit solution satisfiesLimit solution satisfies

Properties of LDC Properties of LDC

Hloc

Hloc

n,hn,H uu

where ΩlocH = common points between

coarse and fine grid


High activity can moveHigh activity can move

Locate high activityLocate high activity Measure features of first coarse grid solution at tMeasure features of first coarse grid solution at tnn

Provide initial values in the new fine grid pointsProvide initial values in the new fine grid points Interpolate in spaceInterpolate in space

AdaptivityAdaptivity

tn-2 tn-1

tn

t


ConservationConservation

Physical problemPhysical problem

if if ··nn = 0 and = 0 and vv··nn = 0 on = 0 on ΩΩ, then , then tracer is conservedtracer is conserved

A A conservative LDCconservative LDC?? Combine LDC with Finite VolumeCombine LDC with Finite Volume DefectDefect Scaling during regriddingScaling during regridding

inPe

1

t2v

0d 0d 0d

FINITE VOLUME ADAPTED LDC ALGORITHM: discrete conservation at convergence of the LDC iteration


Multilevel LDCMultilevel LDC

ttntn-1


A dipole-wall collision problemA dipole-wall collision problem

vv from Navier-Stokes eq. in v- from Navier-Stokes eq. in v- formulation in formulation in ΩΩ = (0,2)x(-1,1) = (0,2)x(-1,1) boundary condition:boundary condition: vv = = 00 on on ΩΩ initial condition:initial condition: a dipole at the center a dipole at the center what happens:what happens: dipole travels, hits the wall, forms vortices dipole travels, hits the wall, forms vortices

(depending on Reynolds number)(depending on Reynolds number) solve by:solve by: spectral method spectral method use:use: external Fortran code external Fortran code

Transport problem in Transport problem in ΩΩ boundary condition:boundary condition: ··nn = 0 on = 0 on ΩΩ initial condition:initial condition: tracer where the dipole hits the wall tracer where the dipole hits the wall what happens:what happens: tracer transported by tracer transported by vv solve by:solve by: FV adapted LDC with 2 levels of refinement & 1 LDC FV adapted LDC with 2 levels of refinement & 1 LDC

iteration/time stepiteration/time step use:use: C++ code C++ code


Implementation of LDCImplementation of LDC

class Problem {const Grid* g;const BoundaryConditions* bc;const InitialCondition* ic;const Defect* def;

public:Problem();void setGrid(const Grid* gg); void setBC(const BoundaryConditions* bbc);void setIC(const InitialCondition* iic);void setDef(const Defect* ddef);int solve();

/* some other stuff */ };

void provideBClocally( const Problem* global, Problem* local);

void computeDefect( Problem* global, const Problem* local );


Numerical results: Re=250, Pe=500Numerical results: Re=250, Pe=500


Numerical results: Re=1250, Pe=2000Numerical results: Re=1250, Pe=2000


ConservationConservation

dydx)t,y,x()t(MTotal quantity of passive tracer

)0(M

)0(M)t(M

LDC might be not fully converged in one iteration


ConclusionsConclusions

LDC is an LDC is an adaptiveadaptive method for solving PDEs method for solving PDEs Coarse and fine grid solution iteratively combinedCoarse and fine grid solution iteratively combined

ExtensionsExtensions of the basic algorithm of the basic algorithm conservative solutionconservative solution multiple levels of refinementmultiple levels of refinement

LDC is applied to transport of passive tracersLDC is applied to transport of passive tracers

casa day 9 may, 2006

Documents