## Communications in Applied Mathematics and Computational Science

- Commun. Appl. Math. Comput. Sci.
- Volume 6, Number 1 (2011), 1-25.

### A high-order finite-volume method for conservation laws on locally refined grids

Peter McCorquodale and Phillip Colella

#### Abstract

We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. The underlying method is a generalization of that developed by Colella, Dorr, Hittinger and Martin (2009) to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combined with fourth-order accurate Runge–Kutta discretization in time. To interpolate boundary conditions at refinement boundaries, we interpolate in time in a manner consistent with the individual stages of the Runge–Kutta method, and interpolate in space by solving a least-squares problem over a neighborhood of each target cell for the coefficients of a cubic polynomial. The method also uses a variation on the extremum-preserving limiter of Colella and Sekora (2008), as well as slope flattening and a fourth-order accurate artificial viscosity for strong shocks. We show that the resulting method is fourth-order accurate for smooth solutions, and is robust in the presence of complex combinations of shocks and smooth flows.

#### Article information

**Source**

Commun. Appl. Math. Comput. Sci., Volume 6, Number 1 (2011), 1-25.

**Dates**

Received: 4 June 2010

Revised: 12 November 2010

Accepted: 28 January 2011

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.camcos/1513732023

**Digital Object Identifier**

doi:10.2140/camcos.2011.6.1

**Mathematical Reviews number (MathSciNet)**

MR2825299

**Zentralblatt MATH identifier**

1252.65163

**Subjects**

Primary: 65M55: Multigrid methods; domain decomposition

**Keywords**

high-order methods finite-volume methods adaptive mesh refinement hyperbolic partial differential equations

#### Citation

McCorquodale, Peter; Colella, Phillip. A high-order finite-volume method for conservation laws on locally refined grids. Commun. Appl. Math. Comput. Sci. 6 (2011), no. 1, 1--25. doi:10.2140/camcos.2011.6.1. https://projecteuclid.org/euclid.camcos/1513732023