Adaptive grid refinement is a technique to speed up the numerical solution of partial differential equations by starting these calculations on a coarse basic grid and refining this grid only there where the solution requires this, e.g. in areas with large gradients. This technique has already been used often, for both steady and unsteady problems. Here, a simple and efficient adaptive grid technique is proposed for the solution of systems of 2D unsteady hyperbolic conservation laws. The technique is applied to the Euler equations of gasdynamics. Extension to other conservation laws or to 3D is expected to be straightforward. A solution algorithm is presented that refines a rectangular basic grid by splitting coarse cells into four, as often as required, and merging these cells again afterwards. The small cells have a shorter time step too, so the grid is refined in space and time. The grid is adapted to the solution several times per coarse time step, therefore the total number of cells is kept low and a fast solution is ensured. The grid is stored in a simple data structure. All grid data are stored in 1D arrays and the grid geometry is determined with, per cell, five pointers to other cells: one `mother' pointer to the cell from which the cell was split and four `neighbour' pointers. The latter are arranged so, that all cells around the considered cell can be quickly found. To determine where the grid is refined, a refinement criterion is used. Three different refinement criteria are studied: one based on the first spatial derivative of the density, one on the second spatial derivative of the density and one on an estimate of the local truncation error, comparable to Richardson extrapolation. Especially the first-derivative \$ ho\$ criterion gives good results. The algorithm is combined with a simple first-order accurate discretisation of the Euler equations, based on Osher's flux function, and tested. A second-order accurate discretisation of the Euler equations is presented that combines a second-order limited discretisation of the fluxes with the time derivatives of the Richtmyer scheme. This scheme can be easily combined with the adaptive-gridding algorithm. Stability is proved for CFL numbers below 0.25. For cells with different sizes, several interpolation techniques are developed, like the use of virtual cells for flux calculation. The scheme is tested with two standard test cases, the 1D Sod problem and the forward-facing step problem, known from the work of Woodward and Colella. The results show that the second-order scheme is more efficient than the first-order scheme. An accuracy, comparable with solutions on uniform grids is obtained, but with at least five times lower computational costs. Results from a last test problem, the shedding of vortices from a flat plate that is suddenly set into motion, confirm that the method can be used for different flow regimes and that it is very useful in practice for analysis of unsteady flow.

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CWI
Modelling, Analysis and Simulation [MAS]
Scientific Computing

Wackers, J. (2003). An adaptive-gridding solution method for the 2D unsteady Euler equations. Modelling, Analysis and Simulation [MAS]. CWI.