In this paper, a centred universal high-order finite volume method for solving hyperbolic balance laws is presented. The scheme belongs to the family of ADER methods where the Generalized Riemann Problems (GRP) is a building block. The solution to these problems is carried through an implicit Taylor series expansion, which allows the scheme to work very well for stiff source terms. The series expansion is high order for the state and requires the evolution in time of spatial derivatives. A Taylor expansion of second order based on a linearization around the resolved state, is proposed for evolving spatial derivatives. A von Neumann stability analysis is carried out to investigate the range of CFL values for which stability and accuracy are balanced. The scheme implements a centred, low dissipation approach for dealing with the advective part of the system which profits from small CFL values. Numerical tests demonstrate that the present scheme can solve, efficiently, hyperbolic balance laws in both conservative and non-conservative form. The scheme is proven to be well-balanced, the C-property, a classical assessment of well-balancing of non-conservative schemes, is numerically demonstrated. An empirical convergence rate assessment shows that the expected theoretical orders of accuracy are achieved up to the fifth order.

本文提出了一种求解双曲平衡定律的中心通用高阶有限体积法。该方案属于 ADER 方法系列，其中广义黎曼问题 (GRP) 是一个构建块。这些问题的解决方案是通过隐式泰勒级数展开来实现的，这使得该方案对于刚性源项非常有效。级数展开是状态的高阶，需要空间导数的时间演化。提出了基于围绕解析状态的线性化的二阶泰勒展开，用于演化空间导数。执行冯诺依曼稳定性分析以研究平衡稳定性和准确性的 CFL 值范围。该计划实施了一个集中的，用于处理系统的平流部分的低耗散方法，该部分受益于小 CFL 值。数值试验表明，本方案可以有效地求解保守和非保守形式的双曲平衡定律。该方案被证明是良好平衡的，C 性质是对非保守方案的良好平衡的经典评估，被数值证明。经验收敛率评估表明，预期的理论精度达到了五阶。对非保守方案的良好平衡的经典评估，以数字方式进行了证明。经验收敛率评估表明，预期的理论精度达到了五阶。对非保守方案的良好平衡的经典评估，以数字方式进行了证明。经验收敛率评估表明，预期的理论精度达到了五阶。