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Time discontinuous Galerkin space-time finite element method (ST/FEM) can be used for developing arbitrary high-order accurate and unconditionally stable time integration schemes for elastodynamics problems. The existing ST/FEMs can be classified as the single-field and two-field ST/FEM: in the former method, either displacement or velocity, is independent and discontinuous in time. In contrast, in the latter method, both displacement and velocity fields are independent and discontinuous in time. Both methods have third-order accuracy for linear interpolation in time, higher than typical time integration schemes used in semi-discretized. However, these methods currently lack a unified computational framework, so each method requires a separate implementation. Therefore, the main goal of the present study is to develop a generalized computational framework that can facilitate the derivation and implementation of the existing linear-in-time ST/FEMs in a unified manner. This framework is developed by realizing that existing methods differ through the treatments of displacement-velocity relationships, which can be unified through displacement functions. In addition, by employing this framework, a new ST/FEM, which is designated as LC v-ST/FEM, is derived from the linear combination of displacement functions of single-field and two-field ST/FEMs. LC v-ST/FEM contains a user-defined parameter α ∈ [ 0 , 1 ] , which can be used for controlling the high-frequency dissipation characteristics. From finite difference analysis and numerical solutions of benchmark problems, it is demonstrated that the proposed method is the third order accurate in time, unconditionally stable, and contains negligible numerical dispersion error for all 0 ≤ α ≤ 1 . Moreover, for α ≠ 0 , the method can attenuate the spurious high-frequency components from the velocity and displacement fields. [ABSTRACT FROM AUTHOR]

In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 퐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L 2 -norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper proposes and rigorously analyzes novel high-order bound-preserving oscillation-eliminating discontinuous Galerkin (BP-OEDG) schemes with the Harten–Lax–van Leer (HLL) numerical flux for the Kapila five-equation two-phase flow model. The evolution equation for the volume fraction is formulated as a conservative advection equation with an additional non-conservative source term. Utilizing a technical splitting approach, we design a uniformly high-order discretization for this source term, incorporating an “upwind” discretization of the non-conservative product at cell interfaces based on the HLL wave speeds. This method inherently ensures the symmetry of the volume fractions, which is crucial for establishing the bound-preserving property. The positivity of partial densities and internal energy is rigorously proven using the geometric quasilinearization (GQL) approach, which transforms the nonlinear pressure positivity constraint into equivalent linear constraints. To suppress potential spurious oscillations, we incorporate a scale-invariant and linearity-invariant oscillation elimination (OE) procedure that damps the DG modal coefficients after each Runge–Kutta stage, as proposed in [M. Peng, Z. Sun and K. Wu, OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws, Math. Comput. (2024), https://doi.org/10.1090/mcom/3998 ]. This OE procedure, acting as a post-processing filter based on the jumps of the DG solution at cell interfaces, is easy to implement, maintains the Abgrall equilibrium condition around an isolated material interface, and preserves the high-order accuracy of the DG schemes. The effectiveness and robustness of the proposed high-order BP-OEDG schemes are demonstrated through several benchmark numerical experiments. [ABSTRACT FROM AUTHOR]

The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. [ABSTRACT FROM AUTHOR]

We propose an interior penalty discontinuous Galerkin (IPDG) method for the Maxwell's transmission eigenvalue problem. First we reformulate the problem as a parameterized eigenvalue problem and define the fixed-point function. The well-posedness of the IPDG discretization for the source problem is proved. Then the uniform convergence of the parameterized discrete solution operator is proved. Finally the error analysis for the IPDG method is realized under reasonable assumptions on the fixed-point function. Numerical examples in 2D and 3D are presented for validation. [ABSTRACT FROM AUTHOR]

We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows us skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. This method is originally proposed in Li et al. (J Sci Comput 80(1):268–288, 2019) that an arbitrarily high-order approximation space with one unknown per element is reconstructed by solving a local least squares fitting problem. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin methods. The least squares problem is modified in this paper, which allows us to establish a norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further inspires us to construct a preconditioner from the piecewise constant space. The preconditioner is shown to be optimal that the upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is independent of the mesh size. In addition, we can enjoy the advantage on the efficiency of the approximation in number of degrees of freedom compared with the standard DG method. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this work, we present a high-order discontinuous Galerkin (DG) method for solving the one-fluid two-temperature Euler equations for non-equilibrium hydrodynamics. In order to achieve optimal order of accuracy as well as suppress potential numerical oscillations behind strong shocks, special jump terms are applied in the DG spatial discretization for the nonconservative equation of electronic internal energy. Moreover, inspired by the solution procedure of Riemann problem, we develop a new HLLC (Harten–Lax–van Leer Contact) approximate Riemann solver for the one-fluid two-temperature Euler equations and use it as a building block for the high-order discontinuous Galerkin method. Several key features of the proposed HLLC approximate Riemann solver are analyzed. Finally, we design typical test cases to numerically verify and demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Preserving the optimal convergence order of discontinuous Galerkin (DG) discretisations in curved domains is a critical and well-known issue. The proposed approach relies on the reconstruction for off-site data (ROD) method developed originally within the finite volume framework. The main advantages are simplicity, since the PDE solver only considers polygonal domains, and versatility, since any type of boundary condition can be imposed. The developed DG–ROD method consists in splitting the boundary conditions treatment and the leading discrete equations from a classical DG formulation into two independent solvers coupled in a simple and efficient iterative procedure. A numerical benchmark is provided to assess the capability of the method with Dirichlet and Neumann boundary conditions prescribed on curved boundaries, demonstrating that the optimal convergence order is effectively achieved. [ABSTRACT FROM AUTHOR]

In this paper, we present the L 2 -norm stability analysis and error estimate for the explicit single-step time-marching discontinuous Galerkin (DG) methods with stage-dependent numerical flux parameters, when solving a linear constant-coefficient hyperbolic equation in one dimension. Two well-known examples of this method include the Runge–Kutta DG method with the downwind treatment for the negative time marching coefficients, as well as the Lax–Wendroff DG method with arbitrary numerical flux parameters to deal with the auxiliary variables. The stability analysis framework is an extension and an application of the matrix transferring process based on the temporal differences of stage solutions, and a new concept, named as the averaged numerical flux parameter, is proposed to reveal the essential upwind mechanism in the fully discrete status. Distinguished from the traditional analysis, we have to present a novel way to obtain the optimal error estimate in both space and time. The main tool is a series of space–time approximation functions for a given spatial function, which preserve the local structure of the fully discrete schemes and the balance of exact evolution under the control of the partial differential equation. Finally some numerical experiments are given to validate the theoretical results proposed in this paper. [ABSTRACT FROM AUTHOR]

In this study, a discontinuous Galerkin (DG) finite element method is employed to solve the nonlinear equilibrium dispersive (ED) model, for the simulation of multi-component gradient elution chromatography using a liquid mobile phase in fixed-bed columns. The ED model comprises a set of coupled nonlinear convection-dominated partial differential equations integrated with nonlinear Langmuir type adsorption isotherms. Gradient elution, characterized by the gradual increase in eluent strength through variations in the chemical composition of the mobile phase, is analyzed. An investigation into the advantages of gradient elution chromatography in comparison to isocratic elution is conducted via a sequence of numerical test experiments that assess the influence of solvent strength, modulator concentration, gradient start and end times, and gradient slope on the elution profiles and temporal moments. It has been observed that gradient elution chromatography influences the behavior, shape, and propagation speed of elution profiles, which subsequently affect the cycle time and column efficiency. The results of this study provide significant insights that are critical for understanding, optimizing, and enhancing gradient elution chromatography. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy. [ABSTRACT FROM AUTHOR]

The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence. These challenges impose significant constraints on computational efficiency, particularly in the domain of engineering applications. To address these concerns, this paper proposes a robust implicit high-order discontinuous Galerkin (DG) method for solving compressible Navier-Stokes (NS) equations on arbitrary grids. The method achieves a favorable equilibrium between computational stability and efficiency. To solve the linear system, an exact Jacobian matrix solving strategy is employed for preconditioning and matrix-vector generation in the generalized minimal residual (GMRES) method. This approach mitigates numerical errors in Jacobian solution during implicit calculations and facilitates the implementation of an adaptive Courant-Friedrichs-Lewy (CFL) number increasing strategy, with the aim of improving convergence and robustness. To further enhance the applicability of the proposed method for intricate grid distortions, all simulations are performed in the reference domain. This practice significantly improves the reversibility of the mass matrix in implicit calculations. A comprehensive analysis of various parameters influencing computational stability and efficiency is conducted, including CFL number, Krylov subspace size, and GMRES convergence criteria. The computed results from a series of numerical test cases demonstrate the promising results achieved by combining the DG method, GMRES solver, exact Jacobian matrix, adaptive CFL number, and reference domain calculations in terms of robustness, convergence, and accuracy. These analysis results can serve as a reference for implicit computation in high-order calculations. [ABSTRACT FROM AUTHOR]

This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection‐diffusion‐reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection‐)diffusion‐reaction problems. [ABSTRACT FROM AUTHOR]

We introduce an innovative postprocessing technique aimed at refining the accuracy of the discontinuous Galerkin method for solving linear delay differential equations (DDEs) with vanishing delays. The fundamental idea behind this postprocessing technique is to enhance the discontinuous Galerkin solution of degree k by incorporating a generalized Jacobi polynomial of degree k + 1 . We demonstrate that this postprocessing step enhances convergence by one order under the L ∞ -norm. Moreover, we apply this technique to both nonlinear DDEs with vanishing delays and linear DDEs with non-vanishing delays. We further validated the theoretical results through a series of numerical examples. [ABSTRACT FROM AUTHOR]

This article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness. [ABSTRACT FROM AUTHOR]

In this article, we present a flux‐based formulation of the hybridizable discontinuous Galerkin (HDG) method for steady‐state diffusion problems and propose a new method derived by letting a stabilization parameter tend to infinity. Assuming an inf‐sup condition, we prove its well‐posedness and error estimates of optimal order. We show that the inf‐sup condition is satisfied by some triangular elements. Numerical results are also provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates. [ABSTRACT FROM AUTHOR]

In this manuscript, we present an innovative approach to address nonlinear Bratu's problem by applying the Discontinuous Galerkin method. Our method accurately computes two-branched numerical solutions of the problem. Depending on the initial guess function, the resulting numerical solution converges either to the upper or to the lower solution of Bratu's problem. We show that our method achieves an optimal convergence rate of order p+1 when p-degree polynomials are used. To confirm the accuracy and efficiency of our proposed method, we present various numerical results. These findings demonstrate the high precision achieved by Discontinuous Galerkin method, enabling the computation of both branches of the solution.

We give a priori error estimates of second order in time fully explicit Runge–Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions τ ≤ ch for piecewise linear and τ ≲ h4/3 for higher order finite elements, we prove a convergence rate for the energy norm ‖⋅‖Lt∞Lx2+|⋅|Lx2Hxλ/2 that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution. [ABSTRACT FROM AUTHOR]

An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform steps, and is simple in algorithmic and software implementations. It was shown that the proposed version of the ADER-DG numerical method is A-stable and L-stable. The ADER-DG numerical method demonstrates superconvergence with convergence order 2 N + 1 for the solution at grid nodes, while the local solution obtained using the local DG predictor has convergence order N + 1 . It was demonstrated that an important applied feature of this implementation of the numerical method is the possibility of using the local solution as a solution with a subgrid resolution, which makes it possible to obtain a detailed solution even on very coarse coordinate grids. The scale of the error of the local solution, when calculating using standard representations of single or double precision floating point numbers, using large values of the degree N, practically does not differ from the error of the solution at the grid nodes. The capabilities of the ADER-DG method for solving stiff ODE systems characterized by extreme stiffness are demonstrated. Estimates of the computational costs of the ADER-DG numerical method are obtained. [ABSTRACT FROM AUTHOR]

The proper solution to the optimal investment decision plays an important role in the decision-making process. Compared to the classical net present value rule, the real option approach captures the value of the flexibility embedded in the investment opportunity. In this paper we study relevant dynamic models interpreting the project as well as flexibility value as the option premium, namely investment projects from the mining industry. Specifically, the problem we face is described by systems of partial differential equations of the Black-Scholes type in terms of time and output price, equipped with the terminal condition enforced at time instants resulting from the specific timing and type of the flexibility that such an investment provides. As a result of that, the comprehensive methodological concept, based on the discontinuous Galerkin approach, is proposed to improve the numerical valuation process. The resulting numerical procedure is sufficiently robust to cope with an early exercise constraint as well as a wide range of model parameters. Finally, the performance of the solving procedure is accompanied within the conceptual case study arising from mining industry, supplemented by comments of practical importance. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical investigation into elucidating the complex dynamics of Richtmyer–Meshkov (RM) phenomena initiated by the interaction of shock waves with forward-triangular light gas bubbles. The triangular bubble is filled with neon, helium, or hydrogen gas, and is surrounded by nitrogen gas. Three different shock Mach numbers are considered: M s = 1.12 , 1.21 , and 1.41. For the numerical simulations, a two-dimensional system of compressible Euler equations for two-component gas flows is solved by utilizing the high-fidelity explicit modal discontinuous Galerkin technique. For validation, the numerical results are compared with the existing experimental results and are found to be in good agreement. The numerical model explores the impact of the Atwood number on the underlying mechanisms of the shock-induced forward-triangle bubble, encompassing aspects such as flow evolution, wave characteristics, jet formation, generation of vorticity, interface features, and integral diagnostics. Furthermore, the impacts of shock strengths and positive Atwood numbers on the flow evolution are also analyzed. Insights gained from this numerical perspective enhance our understanding of RM phenomena triggered by forward-triangular light gas bubbles, with implications for diverse applications in engineering, astrophysics, and fusion research. [ABSTRACT FROM AUTHOR]

In this work, we analyze a discontinuous Galerkin finite element method for the equations of motion that arise in the 2D Oldroyd model of order one. We investigate the existence and uniqueness of semidiscrete discontinuous solutions, as well as the consistency of the scheme. We derive new a priori and regularity results for the discrete solution and establish optimal error estimates in L∞$$ {L}^{\infty } $$‐norm in time and energy norm in space for the velocity and L2$$ {L}^2 $$‐norm in both time and space for the pressure. Uniform estimates are derived for sufficiently small data. We next apply the backward Euler method to the semidiscrete formulation and establish optimal fully discrete error estimates. At the end, we conduct numerical experiments to support our theoretical results and analyze the findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical approximations of the Cahn-Hilliard-Darcy-Stokes system, which is a combination of the modified Cahn-Hilliard equation with the Darcy-Stokes equation. A novel discontinuous Galerkin pressure-correction scheme is proposed for solving the coupled system, which can achieve the desired level of linear, fully decoupled, and unconditionally energy stable. The developed scheme here is implemented by combining several effective techniques, including by adding an additional stabilization term artificially in Cahn-Hilliard equation for balancing the explicit treatment of the coupling term, the stabilizing strategy for the nonlinear energy potential, and a rotational pressure-correction scheme for the Darcy-Stokes equation. We rigorously prove the unique solvability, unconditional energy stability, and optimal error estimates of the proposed scheme. Finally, a number of numerical examples are provided to demonstrate numerically the efficiency of the present formulation. [ABSTRACT FROM AUTHOR]

The limitations of the coverage range and density of transmission wave often result in less-than-ideal results in traveltime tomography. By contrast, joint transmission-reflection traveltime tomography can not only recover deep structures that transmission tomography cannot detect, but also optimize its inversion results. In this article, we perform joint tomography on borehole seismic (vertical seismic profile, reverse vertical seismic profile, and crosswell seismic) data to obtain near-wellbore structures. In the first part, we solve the factored eikonal equation by the discontinuous Galerkin (DG) method to calculate the transmission/reflection traveltime. Owing to the large wavefront curvature near the source point, the traveltime errors generated by the numerical simulation will propagate from the source to all the calculation domains. According to the factorization principle, the equation solution is decomposed into two parts to solve the point-source singularity. To further improve the accuracy of solving traveltime, we use the DG method to solve the factored eikonal equation with additive factors (the factored DG method), obtaining second-order accuracy solution. The adjoint-state method is employed in the inversion section to calculate the gradient of the misfit function. We use the traveltime difference observed inside the model to define the misfit function, which is more suitable for borehole seismic and avoids the influence of surface normal vectors on gradients. Numerical tests applied on models indicate that the joint tomography method has the potential to accurately inverse the seismic structure information near the well and recover the deep underground structure. [ABSTRACT FROM AUTHOR]

This paper presents a gradient-based approach for solving a switching time control problem under the two-phase porous media flow within petroleum extraction. The primary goal is to find an optimal switching time strategy that will maximize the net present value over a predetermined production period. To achieve this aim, a combination of Lagrangian techniques and variational methods is used to handle the proposed optimal control system, followed by the state equations, adjoint equations and the control strategy. The fully implicit discontinuous Galerkin methods are then used to solve the state and adjoint equations in the optimal control problem. Then the optimal switching directions for adjusting the timing of the switching instants to achieve optimality are obtained. The proposed numerical schemes are finally applied to several numerical examples, which further demonstrate that our propose methods are effective and feasible for the purpose of maximizing the net present value. [ABSTRACT FROM AUTHOR]

Residual-type error estimators in energy error for the mortar staggered discontinuous Galerkin (DG) discretizations of second order elliptic equations were developed. The error estimator was proved to be reliable and efficient. An auxiliary function was defined, making it capable of decomposing the energy error into a conforming part and a nonconforming part, which can be combined with the well-known Scott-Zhang local quasi-interpolation operator, and the mortar discrete formulation yields an error estimator in energy error. Importantly, our analysis does not require any saturation assumptions, which are often needed in the literature. Several numerical experiments were presented to confirm our proposed theories. [ABSTRACT FROM AUTHOR]

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In this manuscript, we present an innovative approach to address nonlinear Bratu's problem by applying the Discontinuous Galerkin method. Our method accurately computes two-branched numerical solutions of the problem. Depending on the initial guess function, the resulting numerical solution converges either to the upper or to the lower solution of Bratu's problem. We show that our method achieves an optimal convergence rate of order p + 1 when p -degree polynomials are used. To confirm the accuracy and efficiency of our proposed method, we present various numerical results. These findings demonstrate the high precision achieved by Discontinuous Galerkin method, enabling the computation of both branches of the solution. [ABSTRACT FROM AUTHOR]

This work presents a novel bound-preserving and positivity-preserving direct arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method for compressible multimedium flows by solving the five-equation transport model. The proposed method satisfies the discrete geometric conservation law (D-GCL) which indicates that uniform flow is precisely preserved during the simulation. More importantly, based on the D-GCL condition, we present a theoretical analysis on designing an efficient bound-preserving and positivity-preserving limiting strategy, which is able to maintain the boundedness of the volume fraction and the positivity of the partial density and internal energy, with the aim of avoiding the occurrence of inadmissible solutions and meanwhile improving the computational robustness. The accuracy and robustness of the proposed method are demonstrated by various one- and two-dimensional benchmark test cases. The numerical results verify the well capacity of the proposed high-order ALE-DG method for compressible multimedium flows with both the ideal and stiffened gas equation of state. [ABSTRACT FROM AUTHOR]

For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy by a postprocessing procedure of solving a constrained minimization in each time step. Such a constrained optimization can be formulated as a nonsmooth convex minimization, which can be efficiently solved by first order optimization methods, if using the optimal algorithm parameters. By analyzing the asymptotic linear convergence rate of the generalized Douglas--Rachford splitting method, optimal algorithm parameters can be approximately expressed as a simple function of the number of out-of-bounds cells. We demonstrate the efficiency of this simple choice of algorithm parameters by applying such a limiter to cell averages of a discontinuous Galerkin scheme solving phase field equations for 3D demanding problems. Numerical tests on a sophisticated 3D Cahn--Hilliard--Navier--Stokes system indicate that the limiter is high order accurate, very efficient, and well suited for large-scale simulations. For each time step, it takes at most 20 iterations for the Douglas--Rachford splitting to enforce bounds and conservation up to the round-off error, for which the computational cost is at most 80N with N being the total number of cells. [ABSTRACT FROM AUTHOR]

We present a massively parallel and scalable nodal discontinuous Galerkin finite element method (DGFEM) solver for the time-domain linearized acoustic wave equations. The solver is implemented using the libParanumal finite element framework with extensions to handle curvilinear geometries and frequency dependent boundary conditions of relevance in practical room acoustics. The implementation is benchmarked on heterogeneous multi-device many-core computing architectures, and high performance and scalability are demonstrated for a problem that is considered expensive to solve in practical applications. In a benchmark study, scaling tests show that multi-GPU support gives the ability to simulate large rooms, over a broad frequency range, with realistic boundary conditions, both in terms of computing time and memory requirements. Furthermore, numerical simulations on two non-trivial geometries are presented, a star-shaped room with a dome and an auditorium. Overall, this shows the viability of using a multi-device accelerated DGFEM solver to enable realistic large-scale wave-based room acoustics simulations. [ABSTRACT FROM AUTHOR]

The integral-algebraic equation (IAE) of index 1 is a mixed system of first-kind and second-kind Volterra integral equations (VIEs). In this paper, the discontinuous Galerkin (DG) method is proposed to solve the index-1 IAE, and the optimal global convergence order is obtained. The iterated DG method is introduced in order to improve the numerical accuracy, and the global superconvergence of the iterated DG solution is derived. However, due to the lack of the local superconvergence of the DG residual for first-kind VIEs, there is no local superconvergence for the mixed IAE system of first-kind and second-kind VIEs, and the numerical experiments also verify this. Some numerical experiments are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve the pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin finite element procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach a global system with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the stability and optimal error estimates are also derived by the new technique. Meanwhile, numerical examples are provided to confirm the corresponding theoretical analysis. [ABSTRACT FROM AUTHOR]

We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented. [ABSTRACT FROM AUTHOR]

In this study, we consider a class of non-autonomous time-fractional partial advection-diffusion-reaction equations with Caputo-type fractional derivative. To obtain the numerical solution to the model problem, we apply the non-symmetric interior penalty Galerkin method in space on a uniform mesh and the L1-scheme in time on a graded mesh. It is demonstrated that the computed solution is discretely stable. Superconvergence of error estimates for the proposed method is obtained using the discrete energy-norm. Also, we have applied the proposed method to solve semi-linear problems after linearizing by Newton's linearization process. The theoretical results are verified through numerical experiments. [ABSTRACT FROM AUTHOR]

Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one from among the infinite many weak solutions. Recently, several high order discontinuous Galerkin (DG) methods satisfying entropy inequalities were proposed; see [T. Chen and C.-W. Shu, J. Comput. Phys., 345 (2017), pp. 427-461; J. Chan, J. Comput. Phys., 362 (2018), pp. 346-374; T. Chen and C.-W. Shu, CSIAM Trans. Appl. Math., 1 (2020), pp. 1-52] and the references therein. However, high order numerical methods typically generate spurious oscillations in the presence of shock discontinuities. In this paper, we construct a high order entropy stable essentially oscillation-free DG (OFDG) method for hyperbolic conservation laws. With some suitable modification on the high order damping term introduced in [J. Lu, Y. Liu, and C.-W. Shu, SIAM J. Numer. Anal., 59 (2021), pp. 1299-1324; Y. Liu, J. Lu, and C.-W. Shu, SIAM J. Sci. Comput., 44 (2022), pp. A230-A259], we are able to construct an OFDG scheme with dissipative entropy. It is challenging to make the damping term compatible with the current entropy stable DG framework, that is, the damping term should be dissipative for any given entropy function without compromising high order accuracy. The key ingredient is to utilize the convexity of the entropy function and the orthogonality of the projection. Then the proposed method maintains the same properties of conservation, error estimates, and entropy dissipation as the original entropy stable DG method. Extensive numerical experiments are presented to validate the theoretical findings and the capability of controlling spurious oscillations of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper is devoted to a discontinuous Galerkin (DG) method for nonlinear quasistatic poroelasticity problems. The fully implicit nonlinear numerical scheme is constructed by utilizing DG method for the spatial approximation and the backward Euler method for the temporal discretization. The existence and uniqueness of the numerical solution is proved. Then we derive the optimal convergence order estimates in a discrete H¹ norm for the displacement and in H¹ and L² norms for the pressure. Finally, numerical experiments are supplied to validate the theoretical error estimates of our proposed method. [ABSTRACT FROM AUTHOR]

Summary: We present a novel and simple yet intuitive approach to the stabilization problem for the numerically solved Euler equations with gravity source term relying on a low‐order nodal Discontinuous Galerkin Method (DGM). Instead of assuming isothermal or polytropic solutions, we only take a hydrostatic balance as a given property of the flow and use the hydrostatic equation to calculate a hydrostatic pressure reconstruction that replaces the gravity source term. We compare two environments that both solve the Euler equations using the DGM: deal.II and StormFlash. We utilize StormFlash as it allows for the use of the novel stabilization method. Without stabilization, StormFlash does not yield results that resemble correct physical behavior while the results with stabilization for StormFlash, as well as deal.II model the fluid flow more accurately. Convergence rates for deal.II do not match the expected order while the convergence rates for StormFlash with the stabilization scheme (with the exceptions for the L2$$ {}_2 $$ errors for momentum) meet the expectation. The results from StormFlash with stabilization also fit reference solutions from the literature much better than those from deal.II. We conclude that this novel scheme is a low cost approach to stabilize the Euler equations while not limiting the flow in any way other than it being in hydrostatic balance. [ABSTRACT FROM AUTHOR]

An effective approach based on the discontinuous Galerkin method (DGM) of a high-order accuracy for calculating the initial stage of an aircraft wing's icing is presented. The problem is solved in the Euler approximation for small water droplets that do not affect the main flow. Systems of Navier–Stokes (NS) equations and Euler model equations for the liquid water content and some relations of the ice growth thermodynamics equations are written. The initial and boundary conditions are formulated. A supercomputer DGM implementation is proposed to solve these systems of equations. The efficiency of the parallel version for the code is investigated. Comments are given on the peculiarities of organizing the calculation procedure. The accuracy of the calculation using the DGM schemes of different accuracy orders is investigated. Test cases on the finely dispersed flow of supercooled droplets around a cylinder and a NACA0012 profile are presented. The numerical and experimental data are compared. A conclusion is drawn about the possibility of applying the developed methodology in practice. [ABSTRACT FROM AUTHOR]

In this paper, the sixth-order oscillation-free Hermite weighted essentially non-oscillatory (OF-HWENO) scheme is proposed for hyperbolic conservation laws on structured meshes, where the zeroth- and first-order moments are the variables for the governing equations. The main difference from other HWENO schemes existed in the literature is that we add high-order numerical damping terms in the first-order moment equations to control spurious oscillations for the OF-HWENO scheme. The OF-HWENO scheme not only can achieve the designed optimal numerical order, but also can be easily implemented as we use only one set of stencil in the reconstruction procedure and the same reconstructed polynomials are applied for the zeroth- and first-order moments equations. In order to obtain the adaptive order resolution when facing the discontinuities, a transition polynomial is added in the reconstruction, where the associated linear weights can also be any positive numbers as long as their summation equals one. In addition, the OF-HWENO scheme still keeps the compactness as only immediate neighbor values are needed in the space discretization. Some benchmark numerical tests are performed to illustrate the high-order accuracy, high resolution and robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We study analytical properties of a semi-discrete discontinuous Galerkin (DG) scheme for the kinetic Cucker–Smale (CS) equation. The kinetic CS equation appears in the mean-field limit of the particle CS model and it corresponds to the dissipative Vlasov type equation approximating the large particle CS system. For this proposed DG scheme, we show that it exhibits analytical properties such as the conservation of mass, L 2 -stability and convergence to the sufficiently regular solution, as the mesh-size tends to zero. In particular, we verify that the convergence rate of the DG numerical solution to the sufficiently regular kinetic solution is dependent on the Sobolev regularity of the kinetic soluiton. We also present several numerical simulations for low-dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper, the conditions under which low subsonic torsional flutter in rotor blades occurs are studied experimentally and numerically using the energy method with the assumption of travelling wave modes. As the test geometry, a blade cascade of five blades is considered. The three middle blades have a rotational degree of freedom, and the two outer blades are stationary. The experiments were performed in a wind tunnel at the Institute of Thermomechanics of the Czech Academy of Sciences. The numerical investigation was carried out using the developed CFD solver based on the discretization of the Favre-averaged Navier-Stokes equations by the discontinuous Galerkin method. A novel mesh-deformation algorithm suitable for fluid flow problems with multiple independently moving bodies is proposed. The CFD solver is benchmarked on test problems of flow around stationary and moving aerofoils. Following the energy method, the middle blades are forced to perform a harmonic pitching motion with various interblade phase angles. The total work per cycle of aerodynamic forces acting on the middle blade is evaluated. This analysis is performed both experimentally and numerically. The experiment measurement and the CFD solver predicted the formation of flutter for the same interblade phase angle. Moreover, qualitative agreement is evident between the experiment and simulation on the evaluated work per cycle for various interblade phase angles. [ABSTRACT FROM AUTHOR]

In this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient‐type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank–Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration. [ABSTRACT FROM AUTHOR]