Advection Solver Performance with Long Time Steps, and Strategies for Fast and Accurate Numerical Implementation

Jerry Tessendorf
February, 2015
Technical Note
A detailed look at solving advection equations by using a Characteristic Map. Three items in this note: (1) Using the CM as the tool for advection, substeps can be generated by efficiently, in fact logarithmically fast; (2) The exact solution for the CM is given, leading to an accurate numerical algorithm for using it. Because the solution involves exponentiated 3X3 matrices, the algorithm is relatively slow although very accurate. In fact, for rigid rotations the algorithm is exact. (3) Several standard tests of advection accuracy are evaluated and the error in several popular advection schemes are quantified by comparing them with the exact solution.
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eWave: Using an Exponential Solver on the iWave Problem

Jerry Tessendorf
March 2014
Technical Note
The iWave approach to simulating surface waves is fast and efficient, but suffers stability and artifact problems. By rephrasing the algorithm to employ two first order diffential equations for the displacement and velocity potential, and using an exponential solver, much more accurate simulations result. Importantly, the solution is free of the stability and artifact issues of the previous approach, and maintains high efficiency and flexibility.
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