Hybrid BSQI-WENO Based Numerical Scheme for Hyperbolic Conservation Laws

TL;DR

This paper introduces a hybrid numerical scheme combining B-spline quasi-interpolation and WENO methods to efficiently solve hyperbolic conservation laws, achieving high accuracy and stability near shocks.

Contribution

The paper develops a novel hybrid BSQI-WENO scheme that adaptively captures shocks and smooth regions, improving accuracy and stability over existing methods.

Findings

Achieves convergence rates of 4 and 6 for cubic and quintic BSQI schemes.

Effectively suppresses spurious oscillations near shocks.

Demonstrates superior accuracy and efficiency compared to WENO5 in numerical tests.

Abstract

In this paper, we intend to use a B-spline quasi-interpolation (BSQI) technique to develop higher order hybrid schemes for conservation laws. As a first step, we develop cubic and quintic B-spline quasi-interpolation based numerical methods for hyperbolic conservation laws in 1 space dimension, and show that they achieve the rate of convergence 4 and 6, respectively. Although the BSQI schemes that we develop are shown to be stable, they produce spurious oscillations in the vicinity of shocks, as expected. In order to suppress the oscillations, we conjugate the BSQI schemes with the fifth order weighted essentially non-oscillatory (WENO5) scheme. We use a weak local truncation based estimate to detect the high gradient regions of the numerical solution. We use this information to capture shocks using WENO scheme, whereas the BSQI based scheme is used in the smooth regions. For the time discretization, we consider a strong stability preserving (SSP) Runge-Kutta method of order three. At the end, we demonstrate the accuracy and the efficiency of the proposed schemes over the WENO5 scheme through numerical experiments.