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**Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations.**
*(English)*
Zbl 1218.65085

Summary: We propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge-Kutta (RK) method in time (WENO-RK) by a high order coupling. 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. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy.

We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes.

In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes.

In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35L65 | Hyperbolic conservation laws |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

### Keywords:

adaptive mesh refinement; multiscale simulations; numerical examples; finite difference method; hyperbolic conservation laws; weighted essentially non-oscillatory method; total variation diminishing Runge-Kutta method
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\textit{C. Shen} et al., J. Comput. Phys. 230, No. 10, 3780--3802 (2011; Zbl 1218.65085)

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