Algorithms of very high space–time orders of accuracy for hyperbolic equations in the semidiscrete WENO–DeC framework

• Increasing the order of accuracy of WENO-DeC schemes to very high values provides computational advantages, such as smaller errors for the same computational resources, even in problems involving discontinuities. • Very high order of accuracy is strictly required in certain tests, such as long-time simulations, in order to capture flow features that are smeared by lower-order methods. • Coupling high-order space discretizations with lower-order time discretizations results in computational efficiency losses. • Differences between results obtained using different numerical fluxes tend to diminish as the order of accuracy increases. • WENO-DeC schemes are robust and capable of handling challenging simulations; however, spurious oscillations necessitate the adoption of additional adaptive techniques to prevent simulation crashes. In this work, we provide a deep investigation of a family of arbitrary high order numerical methods for hyperbolic partial differential equations (PDEs), with particular emphasis on very high order versions, i.e., with order higher than 5. More in detail, within the context of a generic Finite Volume (FV) semidiscretization, we consider Weighted Essentially Non–Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. The goal of this paper is twofold. On the one hand, we want to demonstrate the possibility of utilizing very high order schemes in concrete situations and highlight the related advantages. On the other one, we want to debunk the myth according to which, in the context of numerical resolution of hyperbolic PDEs with very high order spatial discretizations, the adoption of lower order time discretizations, e.g., strong stability preserving (SSP) or linearly strong stability preserving ( ℓ SSP ) Runge–Kutta (RK) schemes, does not affect the overall accuracy of the resulting approach and consequently its computational efficiency. Numerical results are reported for the linear advection equation (LAE) and for the Euler equations of fluid dynamics, showing the advantages and the critical aspects of the adoption of very high order numerical methods. Overall, the results indicate the potential for their use in real–life applications, offering advantages in terms of efficiency, such as requiring shorter computational times to achieve a prescribed error, even in problems involving discontinuities. Furthermore, the results confirm order degradation and efficiency loss when coupling very high order space discretizations with lower order SSPRK time discretizations.