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Typical finite difference methods for complex physics or geometry limit themselves to 1st- or 2nd-order in time and space. The reason is that both stability and complexity increase dramatically with higher-order methods:
- Hyperbolic operators can become more oscillatory (due to dispersive higher-order truncation terms)
- Discrete elliptic operators may be more difficult to invert (because of bigger stencils and conditioning)
- Time integration is less stable (regions of absolute stability are smaller)
- Grid generation must include higher-order derivatives to be numerically correct
Our goal is to develop higher-order methods that easily fit into the existing Chombo AMR framework.
One of the compelling reasons is the extreme effectiveness in combination with AMR. For example, with a fourth-order method, using 2 levels of refinement at a factor of 4, truncation error is reduced by almost 5 orders of magnitude (4^2^)4 = 2^16^. In addition, accuracy at coarser resolutions is typically better, making calculations accessible to smaller machines with fewer processors or memory.
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