A hybrid finite element–finite volume method for conservation laws

• Published in: Applied mathematics and computation Vol. 447; p. 127846
• Main Authors: Abgrall, Rémi, Barsukow, Wasilij
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.06.2023; Elsevier
• Subjects: Active Flux; Conservation laws; High order methods; Mathematics; Active Flux; Conservation laws; 65M08; 65M06; 65M60; High order methods; 76N99; high order methods; conservation laws Mathematics Subject Classification : 65M06
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2023.127846

•arbitrarily high-order accurate numerical method for conservation laws.•based on a continuous approximation of the solution.•the degrees of freedom are point values at cell interfaces and moments of the solution inside the cell.•to lowest (3rd) order this method reduces to the Active Flux method.•the update of the moments is achieved immediately by integrating the conservation law over the cell.•two ways how the point values can be updated in time. We propose an arbitrarily high-order accurate numerical method for conservation laws that is based on a continuous approximation of the solution. The degrees of freedom are point values at cell interfaces and moments of the solution inside the cell. To lowest (3rd) order this method reduces to the Active Flux method. The update of the moments is achieved immediately by integrating the conservation law over the cell, integrating by parts and employing the continuity across cell interfaces. We propose two ways how the point values can be updated in time: either by first deriving a semi-discrete method that uses a finite-difference-type formula to approximate the spatial derivative, and integrating this method e.g. with a Runge-Kutta scheme, or by using a characteristics-based update, which is inspired by the original (fully discrete) Active Flux method. We analyze stability and accuracy of the resulting methods.