A Survey of the Isentropic Euler Vortex Problem using High-Order Methods

Abstract

The flux reconstruction (FR) method offers a simple, efficient, and easy to implement method, and it has been shown to equate to a differential approach to discontinuous Galerkin (DG) methods. The FR method is also accurate to an arbitrary order and the isentropic Euler vortex problem is used here to empirically verify this claim. This problem is widely used in computational fluid dynamics (CFD) to verify the accuracy of a given numerical method due to its simplicity and known exact solution at any given time. While verifying our FR solver, multiple obstacles emerged that prevented us from achieving the expected order of accuracy over short and long amounts of simulation time. It was found that these complications stemmed from a few overlooked details in the original problem definition combined with the FR and DG methods achieving high-accuracy with minimal dissipation. This paper is intended to consolidate the many versions of the vortex problem found in the literature and to highlight some of the consequences if these overlooked details remain neglected.

A variable high-order shock-capturing finite difference method with GP-WENO

“…The problem consists of the advection of an isentropic vortex along the diagonal of a Cartesian computational box with periodic boundary conditions. We set up the problem as presented in [105], where the size of the domain is doubled to be [0, 20] × [0, 20] compared to the original setup in [104] to prevent self-interaction of the vortex across the periodic domain. The problem is run for one period of the advection through the domain until the vortex returns to its initial position, where the solution accuracy can be measured against the initial condition.…”

High-order magnetohydrodynamics for astrophysics with an adaptive mesh refinement discontinuous Galerkin scheme

“…Note that it is important to take a large enough domain, so that the perturbations are negligible at the domain border when setting up initial conditions, otherwise waves will appear at the periodic boundaries. This issue was studied in the context of higher-order Euler codes by Spiegel et al (2015) for the similar isentropic vortex test. Fig.…”

Energy- and Entropy-Stable Multidimensional Summation-by-Parts Discretizations on Non-Conforming Grids

“…In this section, we present numerical results validating the conservation, accuracy, and stability properties of the energy-and entropy-stable schemes on non-conforming affine grids. The two-dimensional unsteady isentropic vortex test case with periodic boundary conditions [42] is used for the convergence and efficiency studies and to demonstrate that the schemes are element-wise conservative. This test case is also used to show that entropy is dissipated and conserved using the entropy-conservative schemes (9) and (12) with and without the dissipative term (13), respectively.…”

“…The use of periodic boundary conditions introduces an error since the flow is not uniform at the boundaries (see Section IV.B. of [42] for more details). This error is negligible in our case, because the domain is relatively large.…”