Well-balanced Discontinuous Galerkin Methods Research Articles

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

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.

Numerical approximation of living-man steady state solutions for blood flow in arteries using a well-balanced discontinuous Galerkin scheme

This study developed a well-balanced discontinuous Galerkin method for the one-dimensional blood flow model capable of preserving zero and non-zero velocity steady state solutions. The strategy used was to incorporate the source term into the flux function to rewrite the law of balance in a conservative form, in terms of global equilibrium variables that remained constant in space and time under steady states. Subsequently, the conservative variables were recalculated using the equilibrium variables. Recently, this concept was applied to shallow water and the Euler equations of gas dynamics. Numerical tests verified the well-balanced property of the designed scheme and its ability to capture small steady-state perturbations on relatively coarse meshes accurately.

A High-Order Velocity-Based Discontinuous Galerkin Scheme for the Shallow Water Equations: Local Conservation, Entropy Stability, Well-Balanced Property, and Positivity Preservation

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas. In this work, we present a novel class of locally conservative, entropy stable and well-balanced discontinuous Galerkin (DG) methods for the nonlinear shallow water equation with a non-flat bottom topography. The major novelty of our work is the use of velocity field as an independent solution unknown in the DG scheme, which is closely related to the entropy variable approach to entropy stable schemes for system of conservation laws proposed by Tadmor (in: Tezduyar, Hughes T (eds) Proceedings of the winter annual meeting of the American Society of Mechanical Engineering 1986) back in 1986, where recall that velocity is part of the entropy variable for the shallow water equations. Due to the use of velocity as an independent solution unknown, no specific numerical quadrature rules are needed to achieve entropy stability of our scheme on general unstructured meshes in two dimensions. The proposed DG semi-discretization is then carefully combined with the classical explicit strong stability preserving Runge–Kutta (SSP–RK) time integrators (Gottlieb et al. in SIAM Rev. 43, 89–112, 2001) to yield a locally conservative, well-balanced, and positivity preserving fully discrete scheme. Here the positivity preservation property is enforced with the help of a simple scaling limiter. In the fully discrete scheme, we re-introduce discharge as an auxiliary unknown variable. In doing so, standard slope limiting procedures can be applied on the conservative variables (water height and discharge) without violating the local conservation property. Here we apply a characteristic-wise TVB limiter (Cockburn and Shu in J Comput Phys 141:199–224, 1998) on the conservative variables using the Fu-Shu troubled cell indicator (Fu and Shu in J Comput Phys 347:305–327, 2017) in each inner stage of the Runge–Kutta time stepping to suppress numerical oscillations. This fully discrete can be readily applied to various SWEs simulations without dry areas where the water height is close to zero. The case with dry areas need further special attention, where the velocity approximation can be unphysically large near cells with a small water height, which may eventually crashes the simulation if no special treatment is used near these cells. Here we propose a simple wetting/drying treatment for the velocity update without violating the local conservation property to enhance the robustness of the overall scheme. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

A well-balanced discontinuous Galerkin method for the shallow water flows on erodible bottom

In this paper, we investigate the shallow water (SW) flow over the erodible layer using a fully coupled mathematical model in two-dimensional (2D) space. A well-balanced discontinuous Galerkin (DG) scheme is proposed for solving the SW equations with sediment transport and bed evolution. To achieve the well-balanced property of the numerical scheme easily, the nonlinear SW equations are first reformulated into a new form by introducing an auxiliary variable. Then the DG method is used to discretize the model, in which the FORCE flux is used. By choosing an appropriate value of the auxiliary variable, we can prove that the numerical method can accurately maintain the steady solution in still water, so it is indeed an equilibrium preserving scheme. The well-balanced property can be extended to any numerical flux which satisfies the consistency. Moreover, we investigate the impact on the numerical results if the appropriate value of auxiliary variable is slightly modified. The effectiveness of the numerical method is finally verified by numerical experiments.

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry

ABSTRACT This paper presents high-order Runge–Kutta (RK) discontinuous Galerkin methods for the Euler–Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components that are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane–Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples – including a toy model of stellar core collapse with a phenomenological equation of state that results in core bounce and shock formation – are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock-capturing capability, and total energy conservation.

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

Well-balanced discontinuous Galerkin methods for the one-dimensional blood flow through arteries model with man-at-eternal-rest and living-man equilibria

The movement of blood flow in arteries can be modeled by a system of conservation laws and has a range of applications in medical contexts. In this paper, we present efficient well-balanced discontinuous Galerkin methods for the one-dimensional blood flow model, which preserve the man-at-eternal-rest (zero velocity) and more general living-man (non-zero velocity) equilibria. Recovery of well-balanced states, decomposition of the numerical solutions into the equilibrium and fluctuation parts, and appropriate source term and numerical flux approximations are the key ideas in designing well-balanced methods. Numerical examples are presented to verify the well-balanced property, high order accuracy, good resolution for both smooth and discontinuous solutions, and the ability to capture nearly equilibrium solutions well. We also test the proposed methods on nearly equilibrium flows with various Shapiro numbers. Man-at-eternal-rest well-balanced methods work well for problems with low Shapiro number, but generate spurious flows when Shapiro number gets larger, while the living-man well-balanced methods perform well for all ranges of Shapiro number.

High Order Still-Water and Moving-Water Equilibria Preserving Discontinuous Galerkin Methods for the Ripa Model

Shallow water equations with horizontal temperature gradients, also known as the Ripa system, are used to model flows when the temperature fluctuations play an important role. These equations admit steady state solutions where the fluxes and source terms balance each other. We present well-balanced discontinuous Galerkin methods for the Ripa model which can preserve the still-water or the general moving-water equilibria. The key ideas are the recovery of well-balanced states, separation of the solution into the equilibrium and fluctuation components, and appropriate approximations of the numerical fluxes and source terms. The same framework is also extended to design well-balanced methods for the constant height and isobaric steady state solutions of the Ripa model. Numerical examples are presented to verify the well-balanced property, high order accuracy, and good resolution for both smooth and discontinuous solutions.

A High-Order Well-Balanced Discontinuous Galerkin Method Based on the Hydrostatic Reconstruction for the Ripa Model

A High-Order Well-Balanced Discontinuous Galerkin Method Based on the Hydrostatic Reconstruction for the Ripa Model

Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes

The classical Saint–Venant shallow water equations on complex geometries have wide applications in many areas including coastal engineering and atmospheric modeling. The main numerical challenge in simulating Saint–Venant equations is to maintain the high order of accuracy and well-balanced property simultaneously. In this paper, we propose a high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations. The technique used to maintain well-balanced property is called constant subtraction and proposed in Yang et al. (J Sci Comput 63:678–698, 2015). Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations. Numerical examples with smooth and discontinuous solutions are provided to demonstrate the performance of our proposed DG methods.

Comment on “A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions”

This comment concerns a mistake in the proving of positivity-preserving property for the well-balanced discontinuous Galerkin methods presented by Meister and Ortleb [Meister A, Ortleb S, A positivity preserving and well-balanced DG scheme using finite volume subcells in almost dry regions, Applied Mathematics and Computation 272 (2016) 259–273]. Furthermore, we provide a correct proof which gives the same result.

High order well-balanced discontinuous Galerkin methods for shallow water flow under temperature fields

In this paper, we present high order discontinuous Galerkin methods for the Ripa model. This model maintains the steady state. To achieve the well-balanced property, we first reformulate the governing equations and propose to construct a novel source term approximation. In addition, we apply a flux modification technique to construct the numerical fluxes. Rigorous numerical analysis as well as extensive one- and two-dimensional numerical results all verify that the proposed methods are well-balanced for the steady state, keep genuinely high order accuracy in smooth regions for smooth solutions, and enjoy good resolutions for discontinuous solutions.

Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water flows in open channels

Abstract We construct and study efficient high order discontinuous Galerkin methods for the shallow water flows in open channels with irregular geometry and a non-flat bottom topography in this paper. The proposed methods are well-balanced for the still water steady state solution, and can preserve the non-negativity of wet cross section numerically. The well-balanced property is obtained via a novel source term separation and discretization. A simple positivity-preserving limiter is employed to provide efficient and robust simulations near the wetting and drying fronts. Numerical examples are performed to verify the well-balanced property, the non-negativity of the wet cross section, and good performance for both continuous and discontinuous solutions.

High order well-balanced discontinuous Galerkin methods for Euler equations at isentropic equilibrium state under gravitational fields

Euler equations under gravitational fields often appear in some interesting astrophysical and atmospheric applications. The Euler equations are coupled with gravitational source term due to the gravity and admit hydrostatic equilibrium state where the flux produced by the pressure gradient is exactly balanced by the gravitational source term. In this paper, we construct high order discontinuous Galerkin methods for the Euler equations under gravitational fields, which are well-balanced for the isentropic type hydrostatic equilibrium state. To maintain the well-balanced property, we first reformulate the governing equations in an equivalent form. Then we propose a novel source term approximation based on a splitting algorithm as well as well-balanced numerical fluxes. Rigorous theoretical analysis and extensive numerical examples all suggest that the proposed methods maintain the hydrostatic equilibrium state up to the machine precision. Moreover, one- and two-dimensional simulations are performed to test the ability of the current methods to capture small perturbation of such equilibrium state, and the genuine high order accuracy in smooth regions.

High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations

In this paper, we introduce high order well-balanced discontinuous Galerkin methods for shallow water equations over non-flat bottom topography, which preserve the lake at rest steady state. To achieve the well-balanced property, we propose to construct the numerical fluxes based on the hydrostatic reconstruction idea and to be in combination with a novel source term approximation as well as a decomposition algorithm. Rigorous theoretical analysis and extensive numerical results all verify that the current methods maintain the well-balanced property. In addition, numerical results also indicate that the resulting methods enjoy the ability to accurately capture small perturbations of the lake at rest steady state and keep the genuine high order accuracy for smooth solutions.

Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the Euler equations with gravitation

Many interesting astrophysical and atmospheric problems involve flows near the hydrostatic equilibrium state where the pressure gradient is balanced by the gravitational force. In this paper, we design high order well-balanced discontinuous Galerkin methods for the Euler equations with gravitation, which can preserve the discrete polytropic and isothermal hydrostatic balance states exactly. To achieve the well-balancedness, we propose to combine the numerical fluxes based on a generalized hydrostatic reconstruction, with an equilibrium state recovery technique and a novel source term approximation. Extensive one- and two-dimensional numerical examples are shown to demonstrate the performance of our well-balanced methods, and comparison with non-well-balanced results is included to illustrate the importance of maintaining the balance between pressure gradient and gravitational force numerically.

A Well-Balanced Discontinuous Galerkin Method for the Blood Flow Model

A Well-Balanced Discontinuous Galerkin Method for the Blood Flow Model

Application of positivity-preserving well-balanced discontinuous Galerkin method in computational hydrology

Abstract The positivity-preserving well-balanced discontinuous Galerkin (DG) method (Xing et al. J Sci Comput 57 , 2013) is employed to solve the shallow water equations on an unstructured triangular mesh and to study their applications in computational hydrology. The grid convergence of the DG method is verified via the steady state oblique hydraulic jump problem. The dam-breaking problems with wet and dry river beds are conducted to demonstrate the positivity-preserving property of the scheme. The tidal bores in an idealized estuary problem are simulated to study the development and evolution of the tidal bores from different amplitudes of incoming tidal waves and topography of the river bed bottom. The numerical experiments above demonstrate that the DG method can be applied successfully to these class of problems in computational hydrology.