Adaptive mesh refinement for discontinuous Galerkin method on quadrilateral non-conforming grids Michal A. Kopera PDE’s on the Sphere 2012

Motivation Cut the number of elements down to a minimum necessary to sufficiently well resolve the problem Tackle problems previously difficult or impossible to solve due to limited computational resources Source: NASA

Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements Non-conforming quad-based DG

level 0 Non-conforming flux computation handled by the DG solver Forest of quad-trees approach

Non-conforming quad-based DG Non-conforming flux computation handled by the DG solver Forest of quad-trees approach level 0 level1

Non-conforming quad-based DG level 0 level1 level 2 Non-conforming flux computation handled by the DG solver Forest of quad-trees approach

Non-conforming quad-based DG Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements

Non-conforming quad-based DG Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements

Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements Non-conforming quad-based DG

Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements

Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements Non-conforming quad-based DG ! !

Non-conforming flux computation handled by the DG solver Forest of quad-trees approach Each parent element always replaced by four children At most 2:1 size ratio of face- neighboring elements Non-conforming quad-based DG

How to compute flux? 1) Scatter data from the parent edge to children edges

How to compute flux? 1) Scatter data from the parent edge to children edges 2) Compute flux on children edges like in a conforming case

How to compute flux? 1) Scatter data from the parent edge to children edges 2) Compute flux on children edges like in a conforming case + 3) Gather fluxes from children edges to the parent edge

How to compute flux? 1) Scatter data from the parent edge to children edges 2) Compute flux on children edges like in a conforming case 3) Gather fluxes from children edges to the parent edge 4) Apply fluxes like in a conforming case

+ How to move data through an interface?

Let us define the space for both parent and child faces: with mappings Expanding variables yields

For each children face we require Substitution of expansions and reorganizing the terms yields

Let + We require that After splitting the integrals, plugging-in extensions, reorganizing and variable change we arrive at:

Refinement criterium

Refinement criterium What are the benefits and costs?

thresholdfront position [m] , , , ,754

Analyzing mountain cases Multi-rate time-stepping CG AMRGPU3D + MPIMultigrid ? Outlook Optimized data structures Shallow water

Shallow Water Equations 2D wave with 2D bathymetry Linear hydrostatic mountain