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 ($3^\text{rd}$) 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.
翻译:我们建议了一种任意高顺序精确的保存法数字方法,该方法以解决方案的连续近似为基础。 自由度是单元格界面的点值和单元格内溶液的瞬间。 将这种方法降为活动通量法。 时间的更新立即通过对单元格的保护法的整合、 部件的整合和跨单元格界面的连续性来实现。 我们建议了两种方法来及时更新点值: 要么首先得出一种半分解法, 这种方法使用一定差异型公式来接近空间衍生物, 并结合这种方法, 例如Runge- Kutta 方案, 或者使用基于特性的更新, 由原始( 完全离散的) 活动通量法启发。 我们分析了由此产生的方法的稳定性和准确性。