Discontinuous Galerkin Methods with Optimal Ordering for Fast Reservoir Simulation on General Grids
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This thesis presents a fast solution strategy for a class of hyperbolic transport equations modelling flow in porous media. The basis of the strategy is discontinuous Galerkin schemes which combined with a numerical flux function creates a one-sided dependency between the elements of the spatial discretisation. We take advantage of the one-sided dependency by viewing the elements and the inter-element fluxes as vertices and edges in a directed graph. With a topological sort of the graph we produce an optimal ordering of the elements, allowing for the discrete global system to be decoupled in a sequence of (non)-linear problems. This way, assembly of the full global system is avoided, reducing implementational complexity, computational costs and memory requirements. The procedure is demonstrated on the time-of-flight equation, a stationary tracer equation, and the saturation equation on triangular and tetrahedral grids.