Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Three-dimensional adaptive mesh refinement for hyperbolic conservation laws
SIAM Journal on Scientific Computing
Journal of Computational Physics
SIAM Journal on Scientific Computing
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Adaptive Mesh Refinement Using Wave-Propagation Algorithms for Hyperbolic Systems
SIAM Journal on Numerical Analysis
A smoothness indicator for adaptive algorithms for hyperbolic systems
Journal of Computational Physics
Journal of Scientific Computing
Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations
Journal of Computational Physics
Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces
Journal of Scientific Computing
Adaptive mesh refinement for hyperbolic partial differential equations
Adaptive mesh refinement for hyperbolic partial differential equations
Anti-diffusive flux corrections for high order finite difference WENO schemes
Journal of Computational Physics
Journal of Computational Physics
A fourth-order accurate local refinement method for Poisson's equation
Journal of Computational Physics
SIAM Journal on Scientific Computing
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
Adjoint-based h-p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations
Journal of Computational Physics
Local adaptive mesh refinement for shock hydrodynamics
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II
Journal of Computational Physics
Journal of Scientific Computing
A hybrid adaptive finite difference method powered by a posteriori error estimation technique
Journal of Computational and Applied Mathematics
Well-Balanced Adaptive Mesh Refinement for shallow water flows
Journal of Computational Physics
Journal of Computational Physics
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In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework [4,5] 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) [18,10] 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.