A meshless quasi-interpolation method for solving hyperbolic conservation laws based on the essentially non-oscillatory reconstruction

Abstract

In this paper, a quasi-interpolation method is proposed for solving hyperbolic conservation laws based on the essentially non-oscillatory (ENO) scheme. The hyperbolic equation is discretized in space with the finite difference ENO method and then the semi-discrete system is integrated by the strong stability preserving Runge–Kutta scheme. However, in the step of ENO reconstruction, the finite difference method is replaced by several quasi-interpolation schemes, including the multiquadric quasi-interpolation, the integral-type multiquadric quasi-interpolation and the cubic B-spline quasi-interpolation. Our quasi-interpolation method is simple and easy to implement since it doesn't need to solve any linear system. Moreover, it is also suitable for nonuniform grids and noisy sampling data. Nonlinear hyperbolic problems we target include one-dimensional and two-dimensional Burger equations and also the Euler equation. Numerical results demonstrate that the proposed quasi-interpolation ENO method is stable and has good accuracy.

Keywords:

• Quasi-interpolation
• essentially non-oscillatory method
• finite difference method
• hyperbolic equation

AMS SUBJECT CLASSIFICATIONS:

• 65M70
• 65M22

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Grant No. 12101310), National Natural Science Foundation of China Key Project (Grant No. 11631015), Natural Science Foundation of Jiangsu Province (Grant No. BK20210315), Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province (2022SJYB0259), 2021 Jiangsu Shuangchuang Talent Program (JSSCBS 20210222).