1. CASA Day
9 May, 2006

2. Local Defect Correction for Time-Dependent Problems
Outline
Transport of passive tracers
physical problem, mathematical model
Local Defect Correction (LDC)
basic method and its properties
extensions (conservation, multiple levels of refinement)
Numerical results
9 May 2006
Local Defect Correction for Time-Dependent Problems

3. Transport of passive tracer
a contaminant that does not influence the dynamics of the flow
Goal:
influence of the flow on the tracer
Applications:
dispersion of pollutants
mixing in chemical reactors
9 May 2006
Local Defect Correction for Time-Dependent Problems

4. Local Defect Correction for Time-Dependent Problems
Mathematical model
 = distribution of passive tracer Pe = Peclet number v = given velocity field (or computed solving Navier-Stokes)
 often has a local high activity
solve transport equation using Local Defect Correction (LDC)
9 May 2006
Local Defect Correction for Time-Dependent Problems

5. Local Defect Correction (LDC)
LDC: adaptive method for PDEs with highly localized properties
A coarse grid solution and a fine grid solution are iteratively combined
 Uniform structured grids H h
9 May 2006
Local Defect Correction for Time-Dependent Problems

6. Local Defect Correction for Time-Dependent Problems
One time step with LDC
Integrate on the coarse grid
Provide boundary conditions locally
Integrate on the local fine grid
Until convergence
Compute a defect at forward time
Solve a modified coarse grid problem
Provide new boundary conditions locally
Integrate on the fine grid with updated boundary conditions t
tn-1 Δt tn t tn
tn-1 δt t tn
tn-1 t tn
tn-1
9 May 2006
Local Defect Correction for Time-Dependent Problems

7. Local Defect Correction for Time-Dependent Problems
LDC iteration
Coarse grid
solution at tn
Fine grid
Boundary conditions
Defect
9 May 2006
Local Defect Correction for Time-Dependent Problems

8. Local Defect Correction for Time-Dependent Problems
The defect
PDE
Coarse grid discretization
Fine grid solution is more accurate
Defect
Correction
9 May 2006
Local Defect Correction for Time-Dependent Problems

9. Local Defect Correction for Time-Dependent Problems
Properties of LDC
Convergence
Unconditionally convergent (i.e. for any Δt and H)
One or two iterations suffices
Convergence rate is O(Δt2 H-4) with implicit Euler + centered diff.
Limit solution satisfies
Convergence rate for convection-diffusion equation discretized by implicit Euler + centered differences
where ΩlocH = common points between
coarse and fine grid
9 May 2006
Local Defect Correction for Time-Dependent Problems

10. Local Defect Correction for Time-Dependent Problems
Adaptivity
High activity can move
Locate high activity
Measure features of first coarse grid solution at tn
Provide initial values in the new fine grid points
Interpolate in space
tn-2
tn-1 tn t
9 May 2006
Local Defect Correction for Time-Dependent Problems

11. Local Defect Correction for Time-Dependent Problems
Conservation
Physical problem
if ·n = 0 and v·n = 0 on Ω, then tracer is conserved
A conservative LDC?
Combine LDC with Finite Volume
Defect
Scaling during regridding
FINITE VOLUME ADAPTED LDC ALGORITHM: discrete conservation at convergence of the LDC iteration
9 May 2006
Local Defect Correction for Time-Dependent Problems

12. Local Defect Correction for Time-Dependent Problems
Multilevel LDC t tn
tn-1
9 May 2006
Local Defect Correction for Time-Dependent Problems

13. A dipole-wall collision problem
v from Navier-Stokes eq. in v- formulation in Ω = (0,2)x(-1,1)
boundary condition: v = 0 on Ω
initial condition: a dipole at the center
what happens: dipole travels, hits the wall, forms vortices (depending on Reynolds number)
solve by: spectral method
use: external Fortran code
Transport problem in Ω
boundary condition: ·n = 0 on Ω
initial condition: tracer where the dipole hits the wall
what happens: tracer transported by v
solve by: FV adapted LDC with 2 levels of refinement & 1 LDC iteration/time step
use: C++ code
9 May 2006
Local Defect Correction for Time-Dependent Problems

14. Local Defect Correction for Time-Dependent Problems
Implementation 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 );
9 May 2006
Local Defect Correction for Time-Dependent Problems

15. Numerical results: Re=250, Pe=500
Finite volume with midpoint rule for integrals, centered differences for interpolation. Temporal discretization: implicit Euler at level 0 and 1, Cranc-Nicolson at level 2. Discretization parameters: H = 1/20, Δt = , σ = τ = 5 at every level. Spectral: 256x256 grid, Δtspect=
9 May 2006
Local Defect Correction for Time-Dependent Problems

16. Numerical results: Re=1250, Pe=2000
Finite volume with midpoint rule for integrals, centered differences for interpolation. Temporal discretization: implicit Euler at level 0 and 1, Cranc-Nicolson at level 2. Discretization parameters: H = 1/100, Δt = , σ = τ = 5 at every level. Spectral: 768x768 grid, Δtspect=
9 May 2006
Local Defect Correction for Time-Dependent Problems

17. Local Defect Correction for Time-Dependent Problems
Conservation
Total quantity of passive tracer
LDC might be not
fully converged in
one iteration
9 May 2006
Local Defect Correction for Time-Dependent Problems

18. Local Defect Correction for Time-Dependent Problems
Conclusions
LDC is an adaptive method for solving PDEs
Coarse and fine grid solution iteratively combined
Extensions of the basic algorithm
conservative solution
multiple levels of refinement
LDC is applied to transport of passive tracers
9 May 2006
Local Defect Correction for Time-Dependent Problems