Capturing Near-Equilibrium Solutions: A Comparison between High-Order Discontinuous Galerkin Methods and Well-Balanced Schemes

Abstract

Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-balanced approach. We present a modern numerical implementation of these two strategies and compare them in details, using hydrostatic but also dynamical equilibrium solutions of several simple test cases. Finally, we apply our methodology to the simulation of a protoplanetary disc in centrifugal equilibrium around its star and model its interaction with an embedded planet, illustrating in a realistic application the strength of both methods.

Pressure-induced locking in mixed methods for time-dependent (Navier–)Stokes equations

“…In particular, two cases arise: if classical (nonpressure-robust) elements are used, suboptimal convergence (by two orders for TH-like element families, or one order by equal-order elements) and locking will occur, but if pressure-robust elements are used, optimal L 2 convergence can be maintained. Note that no a-priori knowledge of the equilibrium solution is required, which is a typical disadvantage of well-balanced schemes for hyperbolic conservation laws [5]; the L 2 -orthogonality of certain velocity test functions against arbitrary gradient fields suffices [7].…”

“…definition of the space of discretely divergence-free vector field V 0 h . Similarly, the authors of [5] argue that well-balanced schemes allow to reduce the approximation order of the space discretization in hyperbolic conservation laws.…”

“…Even though it holds for the continuous Helmholtz-Hodge projector P(∇p) = 0, i.e., the divergence-free part of ∇p vanishes exactly [7], the expression (∇p, e h ) may represent a certain consistency error of an appropriate discrete Helmholtz-Hodge projector for non-pressure-robust (i.e., non structure-preserving) space discretizations [7]. Since ∇p balances the sum of all gradient parts in f − u t + ν∆u in the sense of the Helmholtz-Hodge decomposition, different behaviors of different space discretizations reflect their ability to deal with dominant gradient fields in the momentum balance, bulding a connection to certain well-balanced schemes for (vector-valued) hyperbolic conservation laws [5]. Note that ∇(S h (u) − P h (u)) L 2 converges to 0 with the optimal rate for an H 1 norm.…”

“…The short note contributes to the recent scholarly debate on the accuracy of low-order structurepreserving space discretizations, e.g. with respect to the treatment of gradient fields in the momentum balance by well-balanced schemes for the shallow water or compressible Euler equations, and the accuracy of (non-structure-preserving) high-order space discretizations [5]. It demonstrates that the structurepreserving, well-balanced property can be achieved in our setting, if certain discretely divergence-free velocity test functions are even weakly divergence-free in the sense of L 2 [7] -without needing to know the exact form of the equilibrium solution, which is a typical disadvantage of well-balanced schemes for hyperbolic conservation laws [5].…”

Uniformly High-Order Structure-Preserving Discontinuous Galerkin Methods for Euler Equations with Gravitation: Positivity and Well-Balancedness

“…Most of them were proposed for the shallow water equations over a non-flat bottom topology, another prototype example of hyperbolic balance laws; see, e.g., [3,12,19,43,1,37,42,40] and the references therein. In recent years, well-balanced numerical methods for the Euler equations (1) with gravitation have been designed within several different frameworks, including but not limited to the finite volume methods [20,4,15,5,21,16,17,13], gas-kinetic schemes [44,25], finite difference methods [39,10,24], and finite element discontinuous Galerkin (DG) methods [22,6,23,30]. Recently, comparison between high-order DG method and well-balanced DG methods was carried out in [30].…”