Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction

We present a novel approach for solving the shallow water equations using a discontinuous Galerkin spectral element method. The method we propose has three main features. First, it enjoys a discrete well-balanced property, in a spirit similar to the one of e.g. [20]. As in the reference, our scheme does not require any a-priori knowledge of the steady equilibrium, moreover it does not involve the explicit solution of any local auxiliary problem to approximate such equilibrium. The scheme is also arbitrarily high order, and verifies a continuous in time cell entropy equality. The latter becomes an inequality as soon as additional dissipation is added to the method. The method is constructed starting from a global flux approach in which an additional flux term is constructed as the primitive of the source. We show that, in the context of nodal spectral finite elements, this can be translated into a simple modification of the integral of the source term. We prove that, when using Gauss-Lobatto nodal finite elements this modified integration is equivalent at steady state to a high order Gauss collocation method applied to an ODE for the flux. This method is superconvergent at the collocation points, thus providing a discrete well-balanced property very similar in spirit to the one proposed in [20], albeit not needing the explicit computation of a local approximation of the steady state. To control the entropy production, we introduce artificial viscosity corrections at the cell level and incorporate them into the scheme. We provide theoretical and numerical characterizations of the accuracy and equilibrium preservation of these corrections. Through extensive numerical benchmarking, we validate our theoretical predictions, with considerable improvements in accuracy for steady states, as well as enhanced robustness for more complex scenario

Well-balanced finite difference WENO schemes for the blood flow model

The blood flow model maintains the steady state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions

Well-balanced finite volume schemes for hydrodynamic equations with general free energy

Well balanced and free energy dissipative first- and second-order accurate finite volume schemes are proposed for a general class of hydrodynamic systems with linear and nonlinear damping. The natural Liapunov functional of the system, given by its free energy, allows for a characterization of the stationary states by its variation. An analog property at the discrete level enables us to preserve stationary states at machine precision while keeping the dissipation of the discrete free energy. These schemes allow for analysing accurately the stability properties of stationary states in challeging problems such as: phase transitions in collective behavior, generalized Euler-Poisson systems in chemotaxis and astrophysics, and models in dynamic density functional theories; having done a careful validation in a battery of relevant test cases.Comment: Videos from the simulations of this work are available at https://sergioperezresearch.wordpress.com/well-balance

A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context recent implicit-explicit schemes developed in the framework of compressibleflows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases

Implicit and implicit-explicit Lagrange-projection finite volume schemes exactly well-balanced for 1D shallow water system

In this paper we consider the Lagrange-Projection technique in the framework of finite volume schemes applied to the shallow water system. We shall consider two versions of the scheme for the Lagrangian step: one fully implicit and one implicit-explicit, based on how the geometric source term is treated. First and second order well-balanced versions of the schemes are presented, in which the water at rest solutions are preserved. This allows to obtain efficient numerical schemes in low Froude number regimes, as the usual CFL restriction driven by the acoustic waves is avoided.This work is partially supported by projects RTI2018-096064-B-C21 and RTI2018-096064-B-C22 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe”, projects P18-RT-3163 of Junta de Andalucía and UMA18-FEDERJA-161 of Junta de Andalucía-FEDER-University of Málaga. C. Caballero-Cárdenas is supported by the grant FPI2019/087773 funded by MCIN/AEI/10.13039/501100011033 and “ESF Investing in your future”. // Funding for open access charge: Universidad de Málaga/CBUA

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior -- an approximation of the steady solution -- which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions

Fully well balanced entropy controlled DGSEM for shallow water flows: global flux quadrature and cell entropy correction

In this paper we propose a high order DGSEM formulation for balance laws which embeds a general well balanced criterion agnostic of the exact steady state. The construction proposed exploits the idea of a global flux formulation to infer an ad-hoc quadrature strategy called here global flux quadrature. This quadrature approach allows to establish a one to one correspondence, for a given local set of data on a given stencil, between the discretization of a non-local integral operator, and the local steady differential problem. This equivalence is a discrete well balanced notion which allows to construct balanced schemes without explicit knowledge of the steady state, and in particular without the need of solving a local Cauchy problem. The use of Gauss-Lobatto DGSEM allows a natural connection to continuous collocation methods for integral equations. This allows to fully characterize the discrete steady solution with a superconvergence result. The notion of entropy control is also included in the construction via appropriately designed cell artificial viscosity corrections. The accuracy and equilibrium preservation of these corrections are characterized theoretically and numerically. In particular, thorough numerical benchmarking, we confirm all the theoretical expectations showing improvements in accuracy on steady states of one or more orders of magnitude, with a simple modification of a given DGSEM implementation. Robustness on more complex cases involving the propagation of sharp wave fronts is also proved. Preliminary tests on multidimensional problems shows improvements on the preservation of vortex like solutions with important error reductions, despite of the fact that no genuine 2D balancing criterion is embedded