Abstract
Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. The RBF-ENO method is highly flexible in terms of geometry, but its stencil selection algorithm is computational expensive. In this work, we combine the computationally efficient WENO method and the geometrically flexible RBF-ENO method in a hybrid high-resolution essentially nonoscillatory method to solve hyperbolic conservation laws. The scheme is based on overlapping patches with ghost cells, the RBF-ENO method for unstructured patches and a standard WENO method on structured patches. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods to stabilize the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness and flexibility on benchmarks and complex test cases such as the scramjet inflow problem and a conical aerospike nozzle jet simulation.
Highlights
Hyperbolic conservation laws attract substantial interest in science and engineering
We introduce a hybrid high-resolution Essentially nonoscillatory (ENO) method which combines the geometrically flexible radial basis functions (RBFs)-ENO method [21] with the efficient standard two-dimensional weighted ENO (WENO) method [22]
Hybrid high-resolution RBF-ENO method In Section 2.3, we presented the RBF-ENO method which is highly flexible in terms of geometry and ensures high order of accuracy
Summary
Hyperbolic conservation laws attract substantial interest in science and engineering. The interpolation procedure can introduce artificial oscillations which destabilize the scheme Such spurious oscillations, that occur at discontinuities, are a well-known problem for high-order linear methods, referred to as the Gibbs phenomenon [3]. To address this Harten et al [4] proposed the essentially nonoscillatory (ENO) scheme based on the MUSCL approach This method reduces the oscillations that occur due to the interpolation step by choosing the stencil with the least oscillatory behavior, see Fig. 1.
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