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

Preconditioning techniques for Hybridizable discontinuous Galerkin discretizations on GPU architectures

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

Entropy stable high-order discontinuous Galerkin spectral-element methods on curvilinear, hybrid meshes

A time-parallel decoupling method for the four-field Biot’s model with hybridizable discontinuous Galerkin discretization

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 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

This paper introduces the DGFS-BE solver, an open-source Discontinuous Galerkin Fast Spectral solver designed to address the complexities of the Boltzmann equation, a fundamental equation in kinetic theory. The solver combines the Discontinuous Galerkin method for spatial discretization with fast spectral methods for velocity discretization, offering high-order accuracy across various domains. Unlike traditional stochastic methods, DGFS adopts a deterministic approach, avoiding assumptions about the collision kernel and overcoming the limitations of the Direct Simulation Monte Carlo method in rarefied gas flow simulations. The solver’s integration with GPU CUDA technology ensures efficient computation, making it suitable for applications ranging from aerospace engineering to microscale flows. Several test cases, including Couette flow, Fourier conduction, normal shock waves, and pressure-driven microchannel flow, demonstrate the solver’s accuracy and performance. The solver is available at: https://github.com/DGFSproj/ .

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 discontinuous Galerkin pressure correction scheme for the Oldroyd model of order one

This paper presents an enrichment method for the case of re-entrant corner singularities in linear elastic problems but with no enrichment of the finite element solution space. Instead an a posteriori error estimator, with gradient descent, is used to determine the approximate solution of the coefficients for the enrichment functions about each re-entrant corner. The method then approximately removes the singularities from the problem, increasing its regularity. As a result is that exponential convergence of the error can be achieved with uniform refinement in polynomial order. Almost no improvement in the error is expected or observed if uniform refinement in p is used. The approach is termed the Celatus method as the singularities are hidden from view . As problems are made regular it is shown that exponential convergence rates are observed when the Celatus method is combined with h p -adaptivity ( h p -Celatus), requiring far fewer degrees of freedom compared, by orders of magnitude, to traditional finite element analysis with h p -adaptivity. Furthermore each term for the enrichment functions for every re-entrant corner can be evaluated independently. Therefore the method can be implemented in an inherently parallel way. The proposed approach offers an ≈ 10 times reduction in computation time for the same accuracy compared to standard finite element analysis with h p -adaptivity. Additionally, since the solution space is not enriched and is always polynomial, the issue of having near singular matrices does not exist for the Celatus method. The discontinuous Galerkin finite element method is used here, but all equations and methodology are equally applicable to the continuous Galerkin method. • Enrichment of solution achieved without enriching the finite element solution space. • Each corner’s enrichment terms are evaluated independently for parallel execution. • Requires orders of magnitude fewer degrees of freedom than traditional hp-adaptive FEM. • Singularities are removed, converting the problem from non-smooth to smooth, providing convergent results with increasing polynomial order only. • Method avoids issues with near-singular matrices common in enriched methods.

thornado+FLASH-X: A Hybrid Discontinuous Galerkin–Implicit-explicit and Finite-volume Framework for Neutrino-radiation Hydrodynamics in Core-collapse Supernovae**This manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish

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.