A Curvilinear Lagrangian discontinuous Galerkin method for resistive magneto-hydrodynamics

Symmetric direct discontinuous Galerkin method for the biharmonic equation with non-homogeneous boundary condition

Higher-degree rectangular immersed finite elements discontinuous Galerkin methods for elliptic interface problems

Combined discontinuous and continuous Galerkin methods for fractured poroelastic media flow on polytopic grids

A high-order discontinuous Galerkin method for the numerical modeling of epileptic seizures

A sharp-interface discontinuous Galerkin method for simulation of two-phase flow of real gases based on implicit shock tracking

Analysis and design of positivity-preserving high-order discontinuous galerkin methods for two-temperature compressible flows

An entropy stable high-order discontinuous Galerkin method on cut meshes

Multi-species Rosenbluth Fokker-Planck collision operator for discontinuous Galerkin method

High-order multiscale hybridizable discontinuous Galerkin method for a class of one-dimensional oscillatory second-order equations

An unstructured block-based adaptive mesh refinement approach for discontinuous Galerkin method

In this paper, we propose and analyze a strongly mass-conservative numerical scheme for the coupled Navier--Stokes and Darcy--Forchheimer system in both two and three spatial dimensions. The two subproblems are coupled through physically relevant interface conditions, including mass conservation, balance of normal forces, and the Beavers--Joseph--Saffman condition. We employ a staggered discontinuous Galerkin method for the Navier-Stokes equations and use standard mixed finite elements for the Darcy-Forchheimer problem. The proposed formulation incorporates the interface conditions directly, without introducing Lagrange multipliers on the interface or artificial numerical fluxes on the mesh skeleton. As a consequence, although discontinuous Galerkin elements are used in the free-flow region, the resulting discrete velocity field is globally $\bm{H}(\mathrm{div})$-conforming across the entire domain. In particular, the incompressibility constraint is satisfied exactly in the free-flow region, thereby yielding strong mass conservation over the entire computational domain. Under a suitable small-data assumption, we establish the well-posedness of the resulting nonlinear discrete system. Owing to the exact preservation of mass conservation, the proposed scheme exhibits a pressure-robust behavior, in the sense that the velocity approximation is insensitive to pressure effects. Numerical experiments are presented to illustrate the stability and robustness of the method, including its performance in regimes involving small viscosity, large pressure, and limited solution regularity.

Convergence of discontinuous galerkin methods for quasiconvex and relaxed variational problems

High order well-balanced and total-energy-conserving local discontinuous Galerkin methods for compressible self-gravitating Euler equations

Optimal error estimates and stability of a local discontinuous Galerkin method for the stochastic two-dimensional KdV equation

A high-order energy-conserving semi-Lagrangian discontinuous Galerkin method for the Vlasov-Ampère system

Optimal error estimates of the local discontinuous Galerkin method and high-order energy stable RK-SAV scheme for the phase field crystal equation

Abstract We develop a conservative and positivity-preserving discontinuous Galerkin (DG) method for the population balance equation (PBE), which models the distribution of particle numbers across particle sizes due to growth, nucleation, aggregation, and breakage. To ensure number conservation in growth and mass conservation in aggregation and breakage, we design a DG scheme that applies standard treatment for growth and nucleation, and introduces a novel discretization for aggregation and breakage. The birth and death terms are discretized in a symmetric double-integral form, evaluated using a common refinement of the integration domain and carefully selected quadrature rules. Beyond conservation, we focus on preserving the positivity of the number density in aggregation–breakage. Since local mass corresponds to the first moment, the classical Zhang–Shu limiter, which preserves the zeroth moment (i.e., cell average), is not directly applicable. We address this by proving the positivity of the first moment on each cell and constructing a moment-conserving limiter that enforces nonnegativity across the domain. To the best of our knowledge, this is the first work to develop a positivity-preserving algorithm that conserves a prescribed moment. The numerical results verify the accuracy, conservation, and robustness of the proposed method.

This paper presents an enhanced formulation of the Fragile Points Method (FPM), a truly meshless approach for efficiently modeling implicit interfaces in two-dimensional differential equations involving non-homogeneous materials. The proposed framework eliminates the need for specialized numerical integration techniques and provides a systematic mathematical foundation for solving interface problems. Discontinuities in both primary and secondary variables across interfaces are naturally handled through the inherently discontinuous shape functions of FPM. Unlike conventional Galerkin methods, FPM employs simple, local, pointbased polynomial trial and test functions constructed via a generalized finite difference approach. These discontinuous functions bypass the continuity requirements of standard Galerkin frameworks. To address the resulting inconsistency due to discontinuities, we incorporate numerical flux corrections inspired by the discontinuous Galerkin method. The proposed method is validated through several benchmark problems, demonstrating its efficiency and robustness.

This paper develops a local discontinuous Galerkin (LDG) method based on generalized numerical fluxes for solving traveling wave solutions of the modified Buckley-Leverett equation. To achieve efficient computation, the original equation is reformulated into a first-order system by introducing auxiliary variables, followed by spatial discretization using the DG method, while the explicit third-order Runge-Kutta method is adopted for temporal discretization. Based on the antisymmetry of the discrete spatial operator, the stability of the scheme under the energy norm is rigorously established. Numerical experiments demonstrate the robustness of the proposed method in handling convection-dominated problems with Riemann initial data, confirming its capability to accurately capture the shock structures.