CASA Day 9 May, 2006

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

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

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

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

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

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

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

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

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

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

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

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

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

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 = 3 10-4, σ = τ = 5 at every level. Spectral: 256x256 grid, Δtspect= 1.25 10-5 9 May 2006 Local Defect Correction for Time-Dependent Problems

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 = 2 10-5, σ = τ = 5 at every level. Spectral: 768x768 grid, Δtspect= 1.25 10-5 9 May 2006 Local Defect Correction for Time-Dependent Problems

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

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