The arctan/tan and Kepler-Burgers mappings for periodic solutions with a shock, front, or internal boundary layer

Abstract Many periodic solutions have internal regions of rapid change—internal boundary layers. Shock waves and geophysical fronts are one class of examples. A second class is composed of functions which decay rapidly away from a central peak or peaks. Spherical harmonics, Mathieu eigenfunctions, prolate spheroidal wave functions, and geophysical Hough functions may all be locally approximated by Hermite functions (in the appropriate parameter range) and decay exponentially fast outside a narrow subinterval. Similarly, the large amplitude cnoidal waves of the Korteweg-DeVries equation are narrow, isolated peaks which are well approximated by the sech 2 ( y ) form of the solitary wave. In this article, we show that a change-of-coordinate is a powerful tool for resolving such internal boundary layers. In the first part, we develop a general theory of mappings for the spherical harmonic/cnoidal wave class of examples, which decay rapidly away towards the edges of the spatial period. The particular map y = arc tan ( L tan( x )) is a particularly effective choice. Four numerical examples show that this map and the Fourier pseudospectral method are a good team. In the second part, we generalize the earlier theory to describe mappings which asymptote to a constant but non-zero resolution at the ends of the periodicity interval. We explain why the “Kepler-Burgers” mapping is particularly suitable for shock and fronts.

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