RBF Based CWENO Method

Solving hyperbolic conservation laws on general grids can be important to reduce the computational complexity and increase accuracy in many applications. However, the use of non-uniform grids can introduce challenges when using high-order methods. We propose to use a Central WENO (CWENO) scheme based on radial basis function (RBF) interpolation, which is applicable to scattered data. We develop a smoothness indicator, based on RBFs, and CWENO specific weights which depend on the mesh size of the grid to construct an arbitrarily high order RBF-CWENO method. We evaluate the method with multiple examples in one dimension

High-order essentially nonoscillatory methods based on radial basis functions

Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely employed to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose high-order essentially nonoscillatory methods based on radial basis functions (RBFs) to solve hyperbolic conservation laws. We derive a smoothness indicator that guarantees the satisfaction of the sign property of the resulting interpolant on general one-dimensional grids. Based on this algorithm we introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Hence, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. Next, we propose a high-order ENO method based on radial basis functions to solve hyperbolic conservation laws on general two-dimensional unstructured grids. The radial basis function reconstruction offers a flexible framework to deal with ill-conditioned cell arrangements. We define a smoothness indicator based the one-dimensional version and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting to circumvent the stagnation of the error and keep the condition number of the reconstruction bounded. To reduce the computational complexity, we develop the RBF-CWENO method. This method exhibits high-order convergence and robustness when solving challenging problems and is considerably faster. However, the resolution close to shocks and turbulent structures is lower than for the RBF-ENO method. Finally, we present a hybrid high-resolution RBF-ENO method which is based on the RBF-ENO method for unstructured patches and the standard WENO method on structured ones. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods that stabilizes the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness on the scramjet inflow problem and a conical aerospike nozzle jet simulation

Entropy stable essentially nonoscillatory methods based on RBF reconstruction

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids

Hybrid high-resolution RBF-ENO method

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