Yulong Xing

High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206-227; J. Sci. Comput., accepted], we designed a well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, we perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

High-order well-balanced finite volume WENO schemes for shallow water equation with moving water

A characteristic feature of hyperbolic systems of balance laws is the existence of non-trivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Recently a number of so-called well-balanced schemes were developed which satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. In most cases, applications treated equilibria at rest, where the flow velocity vanishes. Here we present a new very high-order accurate, exactly well-balanced finite volume scheme for moving flow equilibria. Numerical experiments show excellent resolution of unperturbed as well as slightly perturbed equilibria.

On the Advantage of Well-Balanced Schemes for Moving-Water Equilibria of the Shallow Water Equations

This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, moving-water well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.

Conservative, discontinuous Galerkin–methods for the generalized Korteweg–de Vries equation

We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg–de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the L2–norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of traveling–wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.

High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms

In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems

Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium

Hyperbolic conservation laws with source terms often admit steady state solutions where the fluxes and source terms balance each other. To capture this balance and near-equilibrium solutions, well-balanced methods have been introduced and performed well in many numerical tests. Shallow water equations have been extensively investigated as a prototype example. In this paper, we develop well-balanced discontinuous Galerkin methods for the shallow water system, which preserve not only the still water at rest steady state, but also the more general moving water equilibrium. The key idea is the recovery of well-balanced states, a special source term approximation, and the approximation of the numerical fluxes based on a generalized hydrostatic reconstruction. We also study the extension of the positivity-preserving limiter presented in [40] in this framework. Numerical examples are provided at the end to verify the well-balanced property and good resolution for smooth and discontinuous solutions.

Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes

The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010)”, the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we explore the extension of these methods on unstructured triangular meshes. The simple positivity-preserving limiter is reformulated, and we prove that the resulting scheme guarantees the positivity of the water depth. Extensive numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.

High Order Well-Balanced WENO Scheme for the Gas Dynamics Equations Under Gravitational Fields

The gas dynamics equations, coupled with a static gravitational field, admit the hydrostatic balance where the flux produced by the pressure is exactly canceled by the gravitational source term. Many astrophysical problems involve the hydrodynamical evolution in a gravitational field, therefore it is essential to correctly capture the effect of gravitational force in the simulations. Improper treatment of the gravitational force can lead to a solution which either oscillates around the equilibrium, or deviates from the equilibrium after a long time run. In this paper we design high order well-balanced finite difference WENO schemes to this system, which can preserve the hydrostatic balance state exactly and at the same time can maintain genuine high order accuracy. Numerical tests are performed to verify high order accuracy, well-balanced property, and good resolution for smooth and discontinuous solutions. The main purpose of the well-balanced schemes designed in this paper is to well resolve small perturbations of the hydrostatic balance state on coarse meshes. The more difficult issue of convergence towards such hydrostatic balance state from an arbitrary initial condition is not addressed in this paper.

New Efficient Sparse Space―Time Algorithms for Superparameterization on Mesoscales

Abstract Superparameterization (SP) is a large-scale modeling system with explicit representation of small-scale and mesoscale processes provided by a cloud-resolving model (CRM) embedded in each column of a large-scale model. New efficient sparse space–time algorithms based on the original idea of SP are presented. The large-scale dynamics are unchanged, but the small-scale model is solved in a reduced spatially periodic domain to save the computation cost following a similar idea applied by one of the authors for aquaplanet simulations. In addition, the time interval of integration of the small-scale model is reduced systematically for the same purpose, which results in a different coupling mechanism between the small- and large-scale models. The new algorithms have been applied to a stringent two-dimensional test suite involving moist convection interacting with shear with regimes ranging from strong free and forced squall lines to dying scattered convection as the shear strength varies. The numerical ...

ABSOLUTELY STABLE LOCAL DISCONTINUOUS GALERKIN METHODS FOR THE HELMHOLTZ EQUATION WITH LARGE WAVE NUMBER

Two local discontinuous Galerkin (LDG) methods using some non- standard numerical uxes are developed for the Helmholtz equation with the rst order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are absolutely stable (hence well- posed) with respect to both the wave number and the mesh size. Optimal order (with respect to the mesh size) error estimates are proved for all wave numbers in the preasymptotic regime. To analyze the proposed LDG methods, they are recasted and treated as (non-conforming) mixed nite element methods. The crux of the analysis is to establish a generalized inf-sup condition, which holds without any mesh constraint, for each LDG method. The generalized inf-sup conditions then easily infer the desired absolute stability of the proposed LDG methods. In return, the stability results not only guarantee the well-posedness of the LDG methods but also play a crucial role in the derivation of the error estimates. Numerical experiments, which conrm the theoretical results and compare the proposed two LDG methods, are also presented in the paper.