# Radial Basis Function ENO and WENO Finite Difference Methods Based on the Optimization of Shape Parameters

@article{Guo2017RadialBF, title={Radial Basis Function ENO and WENO Finite Difference Methods Based on the Optimization of Shape Parameters}, author={Jingyang Guo and Jae-Hun Jung}, journal={Journal of Scientific Computing}, year={2017}, volume={70}, pages={551-575} }

We present adaptive finite difference ENO/WENO methods with infinitely smooth radial basis functions (RBFs). These methods slightly perturb the polynomial reconstruction coefficients with RBFs as the reconstruction basis and enhance accuracy in the smooth region by locally optimizing the shape parameters. Compared to the classical ENO/WENO methods, the RBF-ENO/WENO methods provide more accurate reconstructions and sharper solution profiles near the jump discontinuity. Furthermore the RBF-ENO…

## Figures, Tables, and Topics from this paper

## 17 Citations

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

- Mathematics, Computer ScienceJ. Sci. Comput.
- 2017

We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical…

A RBFWENO finite difference scheme for Hamilton-Jacobi equations

- Computer Science, MathematicsComput. Math. Appl.
- 2020

It is revealed that the proposed schemes in this research prepare more accurate reconstructions and sharper solution near singularities by comparing the RBFENO/RBFWENO schemes and the classical ENO/WenO schemes for some benchmark examples.

On the reconstruction of discontinuous functions using multiquadric RBF-WENO local interpolation techniques

- Computer Science, MathematicsMath. Comput. Simul.
- 2020

This paper proposes a true MQ-RBF–WENO method that does not revert to the classical polynomial WENO approximation near discontinuities, as opposed to what was proposed in Guo and Jung (2017).

A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method

- Mathematics
- 2017

Abstract Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods are efficient high order numerical methods for solving hyperbolic conservation laws designed to reduce the Gibbs…

An improved WENO method based on Gauss-kriging reconstruction with an optimized hyper-parameter

- Computer ScienceJ. Comput. Phys.
- 2020

An adaptive finite-difference WENO method with Gauss-Kriging reconstruction with the potential of improving their accuracy is proposed to reduce dissipation in smooth regions of flow while preserving high-resolution around discontinuities for hyperbolic system of conservation laws.

Adaptive Gaussian radial basis function methods for initial value problems: Construction and comparison with adaptive multiquadric radial basis function methods

- Computer Science, MathematicsJ. Comput. Appl. Math.
- 2021

Various adaptive Gaussian RBF methods for solving IVPs are developed by modifying the classical solvers such as the Euler’s method, midpoint method, Adams–Bashforth method and Adams–Moulton method by replacing the polynomial basis with the GaussianRBFs.

Multiscale RBF-based central high resolution schemes for simulation of generalized thermoelasticity problems

- Computer ScienceFrontiers of Structural and Civil Engineering
- 2018

Average-interpolating radial basis functions (RBFs) are successfully integrated with central high-resolution schemes to achieve a higher-order central method for simulation of generalized coupled thermoelasticity problems including shock (singular) waves in their solutions.

A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation

- MathematicsNumerical Heat Transfer, Part B: Fundamentals
- 2019

Abstract Radial basis functions (RBFs) with multiquadric (MQ) kernel have been commonly used to solve partial differential equation (PDE). The MQ kernel contains a user-defined shape parameter (ε),…

Adaptive Radial Basis Function Methods for Initial Value Problems

- Computer Science, MathematicsJ. Sci. Comput.
- 2020

The aim of this paper is to utilize the radial basis function (RBF) interpolation to modify several finite difference methods and thus enhance the performance in terms of local convergence and find the conditions of the shape parameter that could enhance accuracy.

A numerical study of the local monotone polynomial edge detection for the hybrid WENO method

- Mathematics, Computer ScienceJ. Comput. Appl. Math.
- 2017

A detailed numerical study is provided and it is shown that the monotone polynomial method is efficient and accurate for the 5th order hybrid WENO method.

## References

SHOWING 1-10 OF 29 REFERENCES

A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method

- Mathematics
- 2017

Abstract Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods are efficient high order numerical methods for solving hyperbolic conservation laws designed to reduce the Gibbs…

Non-polynomial ENO and WENO finite volume methods for hyperbolic conservation laws

- Mathematics
- 2016

The essentially non-oscillatory (ENO) method is an efficient high order numerical method for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations, if existent, by adaptively…

Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2010

An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline weighted essentially non-oscillatory (WENO) reconstruction with high order…

A numerical study of the local monotone polynomial edge detection for the hybrid WENO method

- Mathematics, Computer ScienceJ. Comput. Appl. Math.
- 2017

A detailed numerical study is provided and it is shown that the monotone polynomial method is efficient and accurate for the 5th order hybrid WENO method.

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

- Computer Science, MathematicsJ. Comput. Phys.
- 2011

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys…

Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws

- Computer Science, MathematicsJ. Comput. Phys.
- 2007

In this paper, we introduce the multi-domain hybrid Spectral-WENO method aimed at the discontinuous solutions of hyperbolic conservation laws. The main idea is to conjugate the non-oscillatory…

High order Hybrid central-WENO finite difference scheme for conservation laws

- Mathematics
- 2006

In this article we present a high resolution hybrid central finite difference-WENO scheme for the solution of conservation laws, in particular, those related to shock-turbulence interaction problems.…

Two-Dimensional Multi-Domain Hybrid Spectral-WENO Methods for Conservation Laws

- Mathematics
- 2006

Abstract : The multi-domain hybrid Spectral-WENO(Weighted Essentially Non-Oscillatory) method (Hybrid) is introduced for the numerical solution of two dimensional nonlinear hyperbolic systems in a…

An improved weighted essentially non-oscillatory scheme with a new smoothness indicator

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2013

A new smoothness indicator that evaluates the local smoothness of a function inside of a stencil and provides at least the same or improved behavior over the fifth-order WENO-JS scheme, but its advantage seems more salient in two dimensional problems.

Uniformly high order accuracy essentially non-oscillatory schemes III

- Mathematics
- 1987

We continue the construction and the analysis of essentially nonoscillatory shock capturing methods for the approximation of hyperbolic conservation laws. We present an hierarchy of uniformly…