# Andrew Christlieb : A high order adaptive mesh refinement algorithm for hyperbolic conservation laws based on weighted essentially non-oscillatory methods

In this work, we combine the adaptive mesh refinement (AMR) framework with high order finite difference weighted essentially non-oscillatory (WENO) method in space and TVD Runge-Kutta (RK) method in time (WENO-RK) for hyperbolic conservation laws. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. To maintain high order accuracy, we use high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from the coarse to find grid, and at ghost points. The resulting scheme is high order accuracy, robust and efficient, due to the mesh adaptivity and has high order accuracy in both space and time. We have experimented the third and fifth order AMR-finite difference WENO-RK schemes. The accuracy of the scheme is demonstrated by applying the method to several smooth test problems, and the quality and efficiency are demonstrated by applying the method to the shallow water and Euler equations with different challenging initial conditions. From our numerical experiment, we conclude a significant improvement of the fifth order AMR - WENO scheme over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, which we think is due to the very low numerical diffusion of high order schemes. This work is in collaboration with Dr. Chaopeng Shen and Professor Jing-Mei Qiu.

**Category**: Applied Math and Analysis**Duration**: 01:34:49**Date**: March 1, 2010 at 4:25 PM-
**Tags:**seminar, Applied Math And Analysis Seminar

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