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 .
We propose a new family of mapped WENO schemes by using several adaptive control functions and a smoothing approximation of the signum function . The proposed schemes introduce the adaptivity and admit an extensive permitted range of the parameters in the mapping functions . Consequently, they have the capacity to achieve optimal convergence rates , even near critical points. Particularly, one of these new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. It produces numerical solutions with high resolution without generating spurious oscillations, especially for long output times .
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a new family of mapped WENO schemes by using
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, 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
. The proposed schemes introduce the adaptivity and admit an
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, that it enjoys some mathematical advantages in achieving
. Consequently, they have the capacity to achieve
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. Improved numerical results are obtained for one-dimensional linear advection cases, especially over
, even near critical points. Particularly, one of these new schemes with fine-tuned parameters illustrates a significant advantage when solving problems with discontinuities. It produces numerical solutions with high resolution without generating spurious oscillations, especially for
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. 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