A modified adaptive improved mapped WENO method
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We propose several adaptive control functions and a smoothing approximation of the signum function, and apply them to design mapping functions used to construct new mapped weighted essentially non-oscillatory (WENO) schemes. Our method extends the formulation of the mapping functions used in the WENO-IM(k,A) methods developed in the last years. The new method admits a more extensive permitted range of the parameters in the mapping functions, that it enjoys some mathematical advantages in achieving optimal convergence rates. Improved numerical results are obtained for one-dimensional linear advection cases, especially over long output times. Numerical solutions of the 1D Euler equations with the Sod's shock tube problem, the Shu-Osher shock-entropy wave interaction problem and the Woodward-Colella blastwave problem are compared with results in references. Quite satisfied performances using the new method are demonstrated in the simulation of two severe problems as the Titarev-Toro shock-entropy wave interaction problem and the 123 problem.
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