We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time-stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.
We present a high-order entropy stable discontinuous Galerkin (ESDG) method for nonlinear conservation laws on both multi-dimensional domains and on networks constructed from one-dimensional domains. These methods utilize treatments of multi-dimensional interfaces and network junctions which retain entropy stability when coupling together entropy stable discretizations. Numerical experiments verify the stability of the proposed schemes, and comparisons with fully 2D implementations demonstrate the accuracy of each type of junction treatment.
We prove that a class of monotone finite volume schemes for scalar conservation laws with discontinuous flux converge at a rate of $\sqrt{Δx}$ in $\mathrm{L}^1$, whenever the flux is strictly monotone in $u$ and the spatial dependency of the flux is piecewise constant with finitely many discontinuities. We also present numerical experiments to illustrate the main result. To the best of our knowledge, this is the first proof of any type of convergence rate for numerical methods for conservation laws with discontinuous, nonlinear flux. Our proof relies on convergence rates for conservation laws …
In this paper, we develop a family of high order cut discontinuous Galerkin (DG) methods for hyperbolic conservation laws in one space dimension. The ghost penalty stabilization is used to stabilize the scheme for small cut elements. The analysis shows that our proposed methods have similar stability and accuracy properties as the standard DG methods on a regular mesh. We also prove that the cut DG method with piecewise constants in space is total variation diminishing (TVD). We use the strong stability preserving Runge-Kutta method for time discretization and the time step is independent of t…
In this article, we present an extension of the splitting algorithm proposed in [22] to networks of conservation laws with piecewise linear discontinuous flux functions in the unknown. We start with the discussion of a suitable Riemann solver at the junction and then describe a strategy how to use the splitting algorithm on the network. In particular, we focus on two types of junctions, i.e., junctions where the number of outgoing roads does not exceed the number of incoming roads (dispersing type) and junctions with two incoming and one outgoing road (merging type). Finally, numerical example…
In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations \cite{guo2016transport, guo2017adaptive}, a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration of accu…
We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in u and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method -- a numerical method which is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with di…
We study an identification problem which estimates the parameters of the underlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkin method, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertain initial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus on identifying the correct end…
We consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows the flux to change across finitely many points in space whose positions are uncertain. We propose a single- and multilevel Monte Carlo method based on a finite…
In this paper, we present and study discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian partial differential equations. We particularly focus on semi-discrete schemes with spatial discretization only, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin-Bona-Mahony equation, the Camassa-Holm equation, the Korteweg-de Vries equation and the nonli…
A class of high-order discontinuous-Galerkin methods satisfying infinitely many entropy conditions with provable error estimates and strong convergence for general nonlinear conservation laws
In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably inv…
We develop a conservative cut finite element method for an elliptic coupled bulk-interface problem. The method is based on a discontinuous Galerkin framework where stabilization is added in such a way that we retain conservation on macro elements containing one element with a large intersection with the domain and possibly a number of elements with small intersections. We derive error estimates and present confirming numerical results
We design, analyze, and implement a new conservative Discontinuous Galerkin (DG) method for the simulation of solitary wave solutions to the generalized Korteweg-de Vries (KdV) Equation. The key feature of our method is the conservation, at the numerical level, of the mass, energy and Hamiltonian that are conserved by exact solutions of all KdV equations. To our knowledge, this is the first DG method that conserves all these three quantities, a property critical for the accurate long-time evolution of solitary waves. To achieve the desired conservation properties, our novel idea is to introduc…
This article concerns a scalar multidimensional conservation law where the flux is of Panov type and may contain spatial discontinuities. We define a notion of entropy solution and prove that entropy solutions are unique. We propose a Godunov-type finite volume scheme and prove that the Godunov approximations converge to an entropy solution, thus establishing existence of entropy solutions. We also show that our numerical scheme converges at an optimal rate of $\mathcal{O}(\sqrt{\D t}).$ To the best of our knowledge, convergence of the Godunov type methods in multi-dimension and error estimate…
Conditional a priori error estimates of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D
Stabilizing Discontinuous Galerkin Methods Using Dafermos' Entropy Rate Criterion: II -- Systems of Conservation Laws and Entropy Inequality Predictors
A moving discontinuous Galerkin finite element method with interface conservation enforcement (MDG+ICE) is developed for solving the compressible Euler equations. The MDG+ICE method is based on the space-time DG formulation, where both flow field and grid geometry are considered as independent variables and the conservation laws are enforced both on discrete elements and element interfaces. The element conservation laws are solved in the standard discontinuous solution space to determine conservative quantities, while the interface conservation is enforced using a variational formulation in a …
The paper describes the construction of entropy-stable discontinuous Galerkin difference (DGD) discretizations for hyperbolic conservation laws on unstructured grids. The construction takes advantage of existing theory for entropy-stable summation-by-parts (SBP) discretizations. In particular, the paper shows how DGD discretizations -- both linear and nonlinear -- can be constructed by defining the SBP trial and test functions in terms of interpolated DGD degrees of freedom. In the case of entropy-stable discretizations, the entropy variables rather than the conservative variables must be inte…
We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Poisson system written as a hyperbolic system using Hermite polynomials in the velocity variable. These schemes are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system. The proposed scheme employs discontinuous Galerkin discretization for both the Vlasov and the Poisson equations, resulting in a consistent description of th…
We present the two-dimensional unstructured grids extension of the a posteriori local subcell correction of discontinuous Galerkin (DG) schemes introduced in [52]. The technique is based on the reformulation of DG scheme as a finite volume (FV) like method through the definition of some specific numerical fluxes referred to as reconstructed fluxes. High-order DG numerical scheme combined with this new local subcell correction technique ensures positivity preservation of the solution, along with a low oscillatory and sharp shocks representation. The main idea of this correction procedure is to …
A variety of real-world applications are modeled via hyperbolic conservation laws. To account for uncertainties or insufficient measurements, random coefficients may be incorporated. These random fields may depend discontinuously on the state space, e.g., to represent permeability in a heterogeneous or fractured medium. We introduce a suitable admissibility criterion for the resulting stochastic discontinuous-flux conservation law and prove its well-posedness. Therefore, we ensure the pathwise existence and uniqueness of the corresponding deterministic setting and present a novel proof for the…
We propose a limiting procedure to preserve invariant domains with time explicit discrete high-order spectral discontinuous approximate solutions to hyperbolic systems of conservation laws. Provided the scheme is discretely conservative and satisfy geometric conservation laws at the discrete level, we derive a condition on the time step to guaranty that the cell-averaged approximate solution is a convex combination of states in the invariant domain. These states are then used to define local bounds which are then imposed to the full high-order approximate solution within the cell via an a post…
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 [22] 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 so…
In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using a subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. The new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method.…
In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal…
This work compares two Nitsche-type approaches to treat non-conforming triangulations for a high-order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point-to-point interpolation) uses inexact integration with quadrature points prescribed by a primary element. The second approach uses exact integration (mortaring) by choosing quadratures depending on the intersection between non-conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point-to-point interpolation. However, we…
Weighted Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Methods for Conservation Laws
University of Science and Technology of China（中国科学技术大学）；Laoshan Laboratory（崂山实验室）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
