Suresh Nisansala Kodippuli Arachchige

• This work mainly focuses on developing higher-order numerical methods for hyperbolic conservation laws. It introduces two innovative approaches aiming to enhance the accuracy and efficiency of the methods. In the explicit setting, our objective is to modify the Active Flux schemes using the ADER-DG technique. Averaged values of the conserved quantities at the cell centers (cell averages) and the interface values of the conserved quantities (point values) are considered as the degrees of freedom of the classical Active Flux method. The conservative update formula of the proposed higher-order Active Flux (hAF) methods depends on the natural degrees of freedom of the corresponding ADER-DG scheme rather than on the cell averages. We construct a highly accurate local space-time predictor by incorporating the information from point values. The proposed scheme achieves a convergence order of \(N+3\) for degree \(N\) spatial test functions, as it applies information from a more accurate local predictor in the update formula. We propose two new update strategies for the point values. In the first a Riemann problem at each interface is solved using information from local predictor. In the second, more general approach, the conservation law is integrated over time at the cell interfaces. To address non-physical oscillations in hAF schemes near discontinuities or steep gradients, we employ the MOOD limiter with some modifications to the existing one in the literature. For linear hyperbolic problems, we achieve an improvement in the CFL number compared to the ADER-DG schemes. In the implicit context, our approach starts by extending the existing implicit Active Flux schemes to linear systems. Then, we introduce a new update strategy for the point values, creating a new class of higher-order implicit Active Flux methods. The novel Modified Implicit Active Flux (MIAF) schemes can generate various numerical schemes. The performance of the selected MIAF schemes is determined by evaluating how they behave in standard test cases. Additionally, a stability and convergence study has been carried out. Comparing the results of the existing single-step implicit Active Flux schemes with those of the MIAF schemes shows that the new approach yields similar outcomes.