Entropy stable reduced order modeling of nonlinear conservation laws using discontinuous Galerkin methods

We investigate the energy-based discontinuous Galerkin (EDG) methods for solving second-order wave equations. The standard EDG formulation produces spurious oscillations near solution discontinuities and yields incorrect wave speeds when the initial data contains a discontinuity. To address these issues, we introduce an oscillation-free approach, augmented with an additional penalty term, to develop the OF-EDG method. The new formulation effectively suppresses spurious oscillations near discontinuities while preserving high-order accuracy for smooth solutions. We establish stability analysis and provide a priori error estimates for several common numerical flux choices. Through a series of numerical experiments, we demonstrate optimal convergence for smooth solutions and confirm the robustness of the OF-EDG method in maintaining oscillation-free behavior for nonsmooth solutions, both for linear wave equations and those with nonlinear source terms. Furthermore, we highlight the importance of the penalty term for ensuring convergence to the true solution when the initial data contains discontinuities.

An unconditionally stable hybridizable-embedded discontinuous Galerkin method for the phase field crystal equation

An embedded-hybridized discontinuous Galerkin method for the electrohydrodynamics system

In this paper, we concentrate on the superconvergence of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear time-dependent fourth-order equations. The numerical flux for the nonlinear convection term is chosen as the generalized local Lax–Friedrichs flux, and the generalized alternating fluxes are employed for the fourth- and second-order terms, which are beneficial for long time simulations with a slower error growth due to the adjustable numerical viscosities. For nonlinear fourth-order equations with periodic boundary conditions, by using generalized Gauss–Radau projections, a modified projection and correction functions, we show a superconvergent bound for the interpolation errors. Then, by designing the numerical initial condition as an interpolation function of the third-order derivative, we derive supercloseness and thus superconvergence results, no matter whether the wind direction is fixed or not. Specifically, for polynomials of degree k, we obtain (2k+1)th order superconvergence for the numerical flux and cell averages, (k+2)th order superconvergence at generalized Radau points, and (k+1)th order for the error derivative at generalized Radau points, followed by a supercloseness result of order k+2 between the generalized Gauss–Radau projections and the numerical solutions. The superconvergence results are extended to the case with mixed boundary conditions when the wind direction is fixed. A series of numerical examples, including various boundary conditions and nonlinear terms, together with long time simulations, are provided to validate the theoretical results and demonstrate the effectiveness of the method.

ABSTRACT An online adaptive reduced order model (ROM) of the discontinuous Galerkin (DG) method is developed for predicting unsteady scale‐resolved flow simulation. The least‐squares Petrov‐Galerkin (LSPG) projection is chosen as the baseline ROM framework, along with typical hyperreduction techniques for acceleration. Since LSPG requires multiplication operations of the Jacobian and the basis, a Jacobian‐free approach is proposed for forming the low‐dimensional ROM system, to keep consistent with the Jacobian‐free strategy of the original implicit DG method. Then, a comprehensive online adaptation algorithm of the LSPG model is developed by updating the basis and sampling elements with snapshots generated by the full‐order DG solver in an efficient way. The key idea for the adaptation is to update the ROM with the most recent flow information to predict the unseen features. Given a set of parameters, the proposed algorithm firstly runs the full‐order DG solver for a short period, secondly generates the initial basis and reduced mesh, and finally runs the adaptive ROM for future‐state predictions, which enables the model to possess predictive capability. Several benchmark cases have been conducted for verification and comparison to static ROMs. The chosen cases involve typical challenges, that is, transportation and discontinuity, for static ROMs, including the isentropic vortex convection, the Sod shock tube, the Kelvin‐Helmholtz instability, and the two‐dimensional Riemann problem. The results demonstrate that the adaptive ROM is able to effectively address the above challenges encountered by its static counterpart from a predictive perspective while achieving reasonable acceleration.

In adherent cells, actomyosin contractility is regulated mainly by the RhoA signaling pathway, which can be controlled by optogenetics. To model the mechanochemical coupling in such systems, we introduce a finite element framework based on the discontinuous Galerkin method, which allows us to treat cell doublets, chains of cells, and monolayers within the same conceptual framework. While the adherent cell layer is modeled as an actively contracting viscoelastic solid on an elastic foundation, different models are considered for the Rho pathway, starting with a simple linear chain that can be solved analytically and later including direct feedback that can be solved only numerically. Our model predicts signal propagation as a function of coupling strength and viscoelastic timescales and identifies the conditions for optimal cell responses and wave propagation. In general, it provides a systematic understanding of how biochemistry and mechanics simultaneously contribute to the communication of adherent cells.

p-multigrid method for the discontinuous Galerkin discretization of elliptic problems with discontinuous coefficients

A priori error estimates based on lyapunov for the FitzHugh-Nagumo model via interior penalty discontinuous Galerkin method

Hybridizable discontinuous galerkin methods for thermo-poroelastic systems

A nonlinear diffusion acceleration (NDA) scheme is developed to efficiently solve the SN transport equation with anisotropic scattering. The spatial discretization of the high-order transport and low-order diffusion equations is based on the discontinuous Galerkin finite element method. A key innovation of this work is the acceleration of the anisotropic scattering source using the P 1 moment obtained from the diffusion equation discretized with the local discontinuous Galerkin (LDG) method. Numerical experiments on one- and two-dimensional problems are conducted to examine the stability and effectiveness of the proposed LDG-based NDA method. The numerical results show that the proposed method is unconditionally stable and significantly accelerates convergence. The total computational time is reduced compared with the conventional NDA method without anisotropic scattering acceleration.

We construct a non-polynomial local discontinuous Galerkin (LDG) scheme for the prescribed mean curvature equation to approximate boundary gradient blow-up solutions, and obtain error estimates.

Abstract Unsteady phase-field problems such as the Allen-Cahn equation play a central role in the simulation of interface evolution in materials science and multi-phase flows, and at the same time present unique challenges in least-squares discontinuous Galerkin (LS-DG) formulations: time derivatives and numerical traces destroy the standard coercivity of the bilinear form, making rigorous energy stability and error analysis highly nontrivial. To overcome this difficulty, we propose an LS-DG scheme combined with a Crank-Nicolson time discretization for the Allen-Cahn equation. By introducing a stabilization term into the weighted least-squares functional and a trace perturbation term, we recover coercivity in the discontinuous setting and obtain a fully discrete LS-DG scheme for the Allen-Cahn equation with rigorous proofs of unique solvability, discrete energy stability, and optimal error estimates. The proposed method integrates the strengths of discontinuous Galerkin and least-squares finite element methods, and naturally provides a posteriori error estimators for adaptive refinement. We rigorously prove unique solvability, discrete energy stability, and optimal error estimates. Numerical experiments not only confirm the theoretical convergence but also demonstrate the effectiveness of the adaptive refinement strategy and its advantages over classical DG methods in terms of accuracy and efficiency.

High-fidelity neutronic analyses of advanced reactors require deterministic transport solvers capable of handling complex unstructured geometries while maintaining computational efficiency. This work presents the development and verification of three graphics processing unit (GPU)–accelerated deterministic solvers implemented within a unified framework, Neutronics using Deterministic Finite Element Algorithm (NuDEAL): the planar method of characteristics (MOC) coupled with the hybrid finite element method (HFEM), the discontinuous Galerkin method of characteristics (DGMOC), and the discontinuous finite element discrete ordinate method (DFEM-SN). These solvers provide complementary capabilities for consistently solving the multigroup transport equation and can be selectively employed to balance accuracy, computational cost, and memory requirements for a given problem. All the methods emphasize efficient GPU execution by leveraging memory alignment, compressed flux storage, and sequential azimuthal sweeps. The solvers are validated on the C5G7 benchmark and applied to advanced reactor problems, including the Advanced Burner Test Reactor (ABTR), Empire microreactor, and the Molten Salt Reactor Experiment. DFEM-SN achieved the highest accuracy, with eigenvalue errors below 50 pcm, while MOC/HFEM and DGMOC provided superior efficiency, with single-GPU run times comparable to those of large CPU clusters. The results demonstrate that deterministic GPU solvers on unstructured meshes can deliver both accuracy and scalability, enabling practical whole-core simulations for heterogeneous advanced reactors. The unified NuDEAL framework establishes a foundation for future extensions toward transient and multiphysics analyses on large-scale GPU architectures.

Preconditioned JFNK discontinuous Galerkin method for the SN neutron transport equation based on the WINGS framework

Numerical simulations of non-Fourier heat transfer in longitudinal fin using local discontinuous Galerkin methods

Algebraic Multigrid (AMG) methods are state-of-the-art algebraic solvers for Partial Differential Equations. Still, their efficiency depends heavily on the choice of suitable parameters and/or ingredients. Paradigmatic examples include the so-called strong threshold parameter, which controls the algebraic coarse-grid hierarchy, as well as the smoother, i.e. the relaxation methods used on the fine grid to damp out high-frequency components of the error. In AMG, since the coarse grids are constructed algebraically (without geometric intuition), the smoother’s performance is even more critical. For the linear systems stemming from polytopal discretizations, such as Polytopal Discontinuous Galerkin (PolyDG) and Virtual Element (VEM) methods, AMG sensitivity to such choices is even more critical due to the significant variability of the underlying meshes, which results in algebraic systems with different sparsity patterns. In this paper, we focus on the linear systems of equations stemming from polytopal discretizations of second-order elliptic problems. We propose a novel deep learning approach that automatically tunes the strong threshold parameter and the smoother choice in AMG solvers, thereby maximizing AMG performance. We test various differential problems in both two- and three-dimensional settings, with heterogeneous coefficients and polygonal/polyhedral meshes, and demonstrate that the proposed approach generalizes well. In practice, we demonstrate that we can reduce AMG solver time by up to [Formula: see text] with minimal changes to existing PolyDG and VEM software libraries.

A comprehensive numerical investigation of the application of troubled-cells to finite volume methods using a novel monotonicity parameter

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

A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities