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CASA Day 9 May, 2006

CASA Day 9 May, 2006. 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. Transport of passive tracer. Passive tracer:

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CASA Day 9 May, 2006

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  1. CASA Day 9 May, 2006

  2. 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 Local Defect Correction for Time-Dependent Problems

  3. Transport of passive tracer • 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 Local Defect Correction for Time-Dependent Problems

  4. Mathematical model •  = distribution of passive tracerPe = Peclet numberv = given velocity field (or computed solving Navier-Stokes) •  often has a local high activity • solve transport equation using Local Defect Correction (LDC) Local Defect Correction for Time-Dependent Problems

  5. H h 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 Local Defect Correction for Time-Dependent Problems

  6. t tn-1 tn t tn-1 tn t tn-1 tn 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 tn Δt δt Local Defect Correction for Time-Dependent Problems

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

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

  9. 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 where ΩlocH = common points between coarse and fine grid Local Defect Correction for Time-Dependent Problems

  10. t tn tn-2 tn-1 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 Local Defect Correction for Time-Dependent Problems

  11. 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 Local Defect Correction for Time-Dependent Problems

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

  14. 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 ); Local Defect Correction for Time-Dependent Problems

  15. Numerical results: Re=250, Pe=500 Local Defect Correction for Time-Dependent Problems

  16. Numerical results: Re=1250, Pe=2000 Local Defect Correction for Time-Dependent Problems

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

  18. 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 Local Defect Correction for Time-Dependent Problems

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