A novel approach for temporal simulations with very high-order finite volume schemes on polyhedral unstructured grids

Abstract

In the finite volume method (FVM), the temporal term of the equation must be integrated in space with the same accuracy as the other spatial schemes from the governing equation. The classic way to solve this is to consider that the computational value inside the cell presents its mean value. In this work, a new operator is proposed, which converts point values into mean ones enabling simpler high-order spatial schemes from the pointwise framework for unsteady problems. One advantage of these schemes is the low number of contributions to the global matrix when compared to the mean counterpart.In the 2D space, this mean operator uses the weighted least-squares method to compute local polynomials and the Gauss quadrature to integrate cell inertial momentums. The method was verified for fourth, sixth and eighth orders in both Cartesian and unstructured meshes when using the Crank-Nicolson time discretization. Dirichlet and Neumann boundary conditions were also successfully verified. Additionally, the numerical spatial error evolution with the solver runtime and the required memory are studied as efficiency metrics of the implemented high-order schemes. High-order schemes provide faster and more accurate results than the second-order counterpart. To reduce the number of required timesteps, high-order backward differentiation formula (BDF) schemes were verified and they lead to considerable time savings when using high-order spatial schemes.