In this paper, we analyze the local discontinuous Galerkin (LDG) method with generalized numerical fluxes to study the superconvergent properties of one-dimensional linearized KdV equations. Compared with traditional upwind and alternating fluxes, a slower error growth of the LDG solution using generalized numerical fluxes can be obtained for long time simulations. By establishing five energy identities and properties of correction functions with the appropriate numerical initial condition, we derive the supercloseness between the LDG solution and the interpolation function. The errors of the numerical fluxes as well as the cell averages achieve the (2k+ 1)th-order super-convergence. In addition, we prove that the superconvergent rates of the function and derivative values at the interior generalized Radau points are k + 2 and k + 1, respectively. An extension to mixed boundary conditions is given, for which we present the generalized skew-symmetry property and propose an appropriate conservation property for the numerical initial condition. Numerical experiments are shown to demonstrate the theoretical results, including cases with other boundary conditions and nonlinear KdV equations.

In this paper, we develop through a careful selection of the auxiliary variables and numerical fluxes an energy conservative local discontinuous Galerkin (LDG) method based on a hybrid form of the general Euler-Korteweg (EK) equations with a variable capillarity coefficient. This energy conservative LDG discretization is of optimal order of accuracy for alternating numerical fluxes, but not for central numerical fluxes which result in the reduction of one order of accuracy when odd degree polynomial basis functions are used. Also, a relatively simple energy conservative LDG discretization for the EK-equations with an irrotational velocity field is presented. Due to the presence of a highly nonlinear third-order spatial derivative term, which originates from the divergence of the Korteweg stress tensor, we employ the novel semiimplicit spectral deferred correction (SDC) method as temporal discretization. The SDC method can be applied to highly nonlinear ordinary differential equations (ODEs) without separating stiff and non-stiff components and is numerically stable for a time step proportional to the mesh size. Numerical experiments, including ones with adaptive meshes, are performed to illustrate the accuracy and capability of the proposed methods to solve the EK-equations.

This paper presents high-order, well-balanced, path-conservative discontinuous Galerkin (DG) methods for the shallow water linearized moment equations (SWLME), designed to preserve both still and moving water equilibrium states. Unlike the multi-layer shallow water equations, which model vertical velocity variations using multiple distinct layers, the SWLME employs a polynomial expansion of velocity profiles with up to N moments. This approach enables a more detailed representation of vertical momentum transfer and complex velocity profiles while retaining hyperbolicity. However, the presence of non-conservative terms and complex steady-state structures introduces significant numerical challenges. Addressing these challenges, we develop path-conservative DG schemes grounded in the Dal Maso-LeFloch-Murat (DLM) theory for non-conservative products. Our method balances flux gradients, non-conservative terms, and source terms through equilibrium-preserving spaces. For the still water equilibrium, we reformulate the equations into a quasilinear form that eliminates source terms, inherently preserving steady states. For the moving water equilibrium, we extend the DG method by transforming conservative variables into equilibrium variables and employing linear segment paths. Theoretical analysis and numerical experiments demonstrate that the proposed methods achieve exact equilibrium preservation while maintaining high-order accuracy, even in scenarios with vertical velocity variations and complex topographies.

The paper presents a novel, unified analytical framework that establishes strong L^2 -stability for a general class of semi-implicit spectral deferred correction (SISDC) time integrators coupled with discontinuous Galerkin (DG) spatial discretizations for linear partial differential equations. The SISDC method treats the non-stiff term explicitly and the stiff term implicitly. A key contribution is the introduction of refined temporal difference operators and identities specifically designed for SISDC. These new identities are the analytic engine that allows the symmetric implicit spatial operator to control the explicit component, enabling a unified stability analysis. Using this framework, we prove strong L^2 -stability under a time-step restriction τ≤τ _0 , where the constant τ _0 is independent of the spatial mesh size h. The theory applies to a wide family of SISDC schemes and multiple DG spatial discretizations, and numerical experiments are provided to validate the theoretical results.

In this paper, we present a class of decoupled, high-order, low-stage semi-implicit deferred correction (DC) schemes to simulate the phase-field dendritic crystal growth model with anisotropy, which are constructed upon the second-order backward differentiation formula (BDF) scheme. The model comprises a strongly nonlinear system coupling the anisotropic Allen-Cahn model with the thermal diffusion model. In order to linearize nonlinear terms arising from the gradient-dependent anisotropic coefficient, we introduce linear stabilization terms into both the initial BDF prediction and the correction stages within the DC framework. This design yields a linear, decoupled, semi-implicit, and computationally efficient scheme, as each iteration requires solving only a small number of Laplace-type problems with constant coefficients. From a theoretical perspective, we establish a rigorous proof of energy stability for the coupled second-order linear implicit BDF scheme in the isotropic case. Finally, a series of two- and three-dimensional numerical experiments on anisotropic models are conducted to test the convergence rates, energy dissipation behavior, and computational efficiency of the developed decoupled semi-implicit algorithms.

In this paper, we construct high-order stabilization terms for semi-implicit deferred correction methods, focusing on convection-diffusion-reaction problems. Optimal choices for stability parameters are investigated and the upper bounds of the timestep to ensure both stability and high-order accuracy are analyzed. Utilizing Fourier analysis, we apply von Neumann stability and Schur theory to new single- and multi-step deferred correction methods. Numerical experiments validate the analysis of the stability parameter estimates and timestep constraints for these methods.

We propose a kernel compensation mimetic difference (MD) scheme to solve the grad-div eigenvalue problem. This method utilizes a curl-curl type compensation operator along with carefully selected boundary conditions to effectively manage the infinite-dimensional kernel of the grad-div operator. To ensure high accuracy, we apply stencil-based MD operators to discretize the grad-div operator under Dirichlet boundary conditions. This results in a numerical scheme characterized by a sparse stiff matrix with a narrow bandwidth while achieving high-order accuracy. We construct the compensation operator with a proper boundary condition that is orthogonal to the discrete grad-div operator. A generalized identification method for spurious eigenvalues is presented. The resulting scheme offers several advantages, including high-order accuracy, enhanced computational efficiency with reduced memory usage, and excellent scalability for parallel computation. Numerical tests demonstrate that our approach not only converges at the expected rates but also performs satisfactorily in terms of speed.

We analyze the stability of the Hagstrom-Warburton nonreflecting boundary conditions (HW-NRBCs) for the first-order time-dependent Maxwell equations. The HW-NRBCs enjoy very small reflection coefficients and do not use high-order derivatives or nonlocal boundary operators, which makes them well-suited for high-order accurate numerical discretizations. The main result of this paper is an L^2 a-priori bound on the solution in terms of initial and boundary data and volume sources in a half-space. To obtain this result, we first derive a mapping that takes outgoing components to ingoing components of the solution of the Maxwell equations with HW-NRBCs, which shows that the Kreiss condition does not hold uniformly. Next, to prove stability, several symmetrizers are constructed, which establishes well-posedness in a generalized sense.

Subsurface regions critically govern surface events, such as the interactions with reactants in heterogeneous catalysis, thereby significantly modulating catalytic performance. However, precise control of subsurface atomic arrangement remains challenging due to complex metal-adsorbate interactions and limited structural accessibility. Here we achieve precise control of subsurface atomic layer in platinum-based intermetallic compounds through targeted positioning of heterometallic atoms to subsurface via in-situ constructed atomic diffusion pathways. This site-specific placement and subsequent thermodynamic-induced atomic rearrangement are governed by surface energy minimization and adsorbate-induced segregation. Through atomic-precision subsurface engineering, we successfully synthesize a series of L10 (face-centered tetragonal, fct)-PtFe@PtMsub, where Msub represents heteroatoms (Ru, Rh, Pd, Ag) incorporated into subsurface layers. As demonstrated, the as-synthesized L10-PtFe@PtPdsub simultaneously stabilizes ligand and strain effects, thereby breaking the trade-off in L10-PtM with Pt skin, where Pt skin typically quenches ligand effects while introducing strain effects. Consequently, L10-PtFe@PtPdsub/C catalyst demonstrates practical proton exchange membrane fuel cells performance, simultaneously delivering high activity and durability. This work provides a rational strategy for catalyst design that promotes the understanding of subsurface active sites in heterogeneous catalysis.

This paper presents and analyzes energy stable local discontinuous Galerkin (LDG) methods for solving the cubic nonlinear shallow water wave equations, also referred to as the Camassa–Holm–Novikov (CHN) equations. These methods are characterized by their high-order accuracy and effectiveness in addressing the intricate nonlinear terms inherent to the CHN equations. The success of the proposed numerical schemes is due to the careful selection of the numerical fluxes at the cell boundaries, which are crucial for ensuring stability and accuracy. We employ a specific interpolation function to derive optimal a priori error estimates and provide a comprehensive analysis of the quadratic Taylor expansion of the cubic nonlinearity. In addition, our numerical schemes show excellent performance for different peakon solutions. Numerical experiments validate the accuracy and efficiency of the proposed methods, demonstrating their effectiveness in practical applications.

We construct higher order accurate bounds preserving time-implicit Discontinuous Galerkin (DG) discretizations for the reactive Euler equations modelling multispecies and multireaction chemically reactive flows. In numerical discretizations of chemically reactive flows, the time step can be significantly limited because of the large difference between the fluid dynamics time scales and the reaction time scales. In addition, the density and pressure should be nonnegative and the mass fractions between zero and one, which imposes constraints on the numerical solution that must be satisfied to obtain physically reliable solutions. We address these issues using the following steps. Firstly, we develop the Karush-Kuhn-Tucker (KKT) limiter for the chemically reactive Euler equations, which imposes bounds on the numerical solution using Lagrange multipliers, and solve the resulting KKT mixed complementarity problem using a semi-smooth Newton method. The disparity in time scales is addressed using a fractional step method, separating the convection and reaction steps, and the use Harten's subcell resolution technique is used to deal with stiff source terms in chemically reactive flows. Numerical results are shown to demonstrate that the bounds preserving KKT-DIRK-DG discretizations are higher order accurate for smooth solutions and able to capture complicated stiff multispecies and multireaction flows with discontinuities.

In this paper, we introduce a novel well-balanced discontinuous Galerkin (DG) method for the rotating shallow water equations, which is founded on the flux globalization approach. Our method entails the integration of the source term into the global fluxes, thereby establishing a quasi-conservative formulation of the equations. The well-balanced property is maintained by ensuring the equilibrium at the nodes of the Lagrange DG basis and through a tailored treatment of the numerical flux. Furthermore, we employ linear segment paths between equilibrium variables at the cell interface to preserve the equilibrium state of the scheme. This strategy allows us to handle more complex equilibrium states, including those with spatially global integral quantities, and accommodates discontinuous bottom topography. We conduct a comprehensive series of numerical experiments on shallow water models, including those with and without Coriolis forces. These experiments confirm the high-order accuracy of our DG method and its ability to exactly preserve equilibrium for intricate moving steady states. Additionally, the method successfully propagates small perturbations of the steady state with high-resolution and oscillation-free solutions, even in the presence of challenging bottom topography conditions.

In this article, we develop an energy stable local discontinuous Galerkin (LDG) method for the isothermal Navier-Stokes-Korteweg (NSK) equations. Since the test and trial functions in LDG discretisations have to be in the same finite element space, it is difficult to obtain energy stable LDG discretizations for the isothermal NSK equations. To bridge this gap we first write the pressure into a free energy function form and introduce the velocity as a variable, then we use an extra auxiliary variable containing both the free energy function and the square of the velocity. These auxiliary variables are chosen in the stability analysis as test functions for the density and momentum balance equations. Using the Crank-Nicolson (CN) time integration method, we can prove then the stability of the CN-LDG method. Numerical experiments are provided to demonstrate the theoretical results, in particular on adaptive meshes.

BACKGROUND:Proton minibeam radiation therapy (pMBRT) poses computational challenges in achieving accurate dose calculations for sub-millimeter minibeams, necessitating fine spatial resolution (e.g., 0.2 mm) in the dose calculation grid. PURPOSE:This work proposes a spatially adaptive dose calculation method that considers pMBRT beam characteristics, including high-dose gradients and spatial fractionation. The method dynamically adjusts the dose calculation grid resolution for computational efficiency while preserving accuracy, employing strategies such as adaptive mesh refinement and hierarchical dose calculation. METHODS:The proposed spatially-adaptive method addresses the intrinsic characteristics of pMBRT, focusing on sharp dose gradients and spatial fractionation at the beam entrance. Within this adaptive dose calculation framework, the conventional uniform dose grid is replaced by a multiscale adaptive grid, incorporating fine spatial sampling in areas of high peak-to-valley dose ratio (PVDR) to ensure precise dose calculation. The sampling rate gradually decreases as depth increases from the beam entrance, optimizing computational efficiency. A dual-grid inverse optimization method is developed for spatially-adaptive grid-based treatment planning. Validation of the adaptive method involves comparing dose calculation and treatment planning results on an adaptive dose grid ("ADAPTIVE") with those on a fine dose grid ("FINE") with a resolution of 0.2 mm and a coarse dose grid ("COARSE") with a resolution of 3 mm. Both ADAPTIVE and COARSE are interpolated to align with the FINE, facilitating accurate dose comparisons. RESULTS:The efficacy of ADAPTIVE for pMBRT is demonstrated by comparing dose calculation and treatment planning with FINE and COARSE. The dose calculation from ADAPTIVE was highly consistent with that from FINE (e.g., Gamma index passing rate in reference to FINE was 100% for the prostate case), while COARSE (Gamma index passing rate 73%) deviated substantially from FINE. On the other hand, the number of dose calculation grids for ADAPTIVE was only 1/15 of that for FINE, and the computational time for ADAPTIVE was less than one-tenth of that for FINE. In terms of the treatment planning difference, ADAPTIVE provided the same PVDR and almost the same dose distribution (e.g., Gamma index passing rate 100% for the prostate case) as FINE, while COARSE (Gamma index passing rate 64%) again was drastically different from FINE. CONCLUSIONS:This work introduces a novel spatially adaptive high-resolution dose calculation method for pMBRT. The proposed spatially adaptive method efficiently and accurately calculates voxel-by-voxel dose distribution and PVDR for pMBRT, significantly reducing the number of required dose calculation grids, compared to the spatially-uniform method.

In this paper, we develop a general framework for the design of the arbitrary high-order well-balanced discontinuous Galerkin (DG) method for hyperbolic balance laws, including the compressible Euler equations with gravitation and the shallow water equations with horizontal temperature gradients (referred to as the Ripa model). Not only the hydrostatic equilibrium including the more complicated isobaric steady state in Ripa system, but our scheme is also well-balanced for the exact preservation of the moving equilibrium state. The strategy adopted is to approximate the equilibrium variables in the DG piecewise polynomial space, rather than the conservative variables, which is pivotal in the well-balanced property. Our approach provides flexibility in combination with any consistent numerical flux, and it is free of the reference equilibrium state recovery and the special source term treatment. This approach enables the construction of a well-balanced method for non-hydrostatic equilibria in Euler systems. Extensive numerical examples such as moving or isobaric equilibria validate the high order accuracy and exact equilibrium preservation for various flows given by hyperbolic balance laws. With a relatively coarse mesh, it is also possible to capture small perturbations at or close to steady flow without numerical oscillations.

In this paper, we present discontinuous Galerkin (DG) finite element discretizations for a class of linear hyperbolic port-Hamiltonian dynamical systems. The key point in constructing a port-Hamiltonian system is a Stokes-Dirac structure. Instead of following the traditional approach of defining the strong form of the Dirac structure, we define a Dirac structure in weak form, specifically in the input-state-output form. This is implemented within broken Sobolev spaces on a tessellation with polyhedral elements. After that, we state the weak port-Hamiltonian formulation and prove that it relates to a Poisson bracket. In our work, a crucial aspect of constructing the above-mentioned Dirac structure is that we provide a conservative relation between the boundary ports. Next, we state DG discretizations of the port-Hamiltonian system by using the weak form of the Dirac structure and broken polynomial spaces of differential forms, and we provide a priori error estimates for the structure-preserving port-Hamiltonian discontinuous Galerkin (PHDG) discretizations. The accuracy and capability of the methods developed in this paper are demonstrated by presenting several numerical experiments.

In this paper, we propose a coupled discontinuous Galerkin (DG) and continuous Galerkin (CG) scheme for solving the nonlinear evolution Davey-Stewartson (DS) system in dimensionless form. The DS system consists of two coupled nonlinear and complex structure partial differential equations. The wave's amplitude in the first equation is solved by the high-efficiency local DG method, and the velocity in the second equation is obtained by a standard CG method. No matching conditions are needed for the two finite element spaces since the normal component of the velocity is continuous across element boundaries. The main strengths of our approach are that we combine the advantage of DG and CG methods, using DG methods handling the nonlinear Schrodinger equation to obtain high parallelizability and high-order formal accuracy, using the continuous finite elements solving the velocity to maintain total energy conservation. We prove the energy-conserving properties of our scheme and error estimates in L2-norm. However, the non-linearity terms bring a lot of trouble to the proof of error estimates. With the help of energy-conserving properties, we construct a series of energy equations to obtain error estimates. Numerical tests for different types of systems are presented to clarify the effectiveness of numerical methods.

In this paper, we propose an efficient adaptive algorithm for the photon-electron coupled Boltzmann equations in radiation therapy. The algorithm employs adaptive mesh technologies for both angular and spatial discretization and effectively solves the equation across various domains with a reduced number of degrees of freedom and computational time while maintaining accuracy. For spatial discretization, we employ an adaptive discontinuous Galerkin scheme based on material redefinition to automatically conduct adaptive refinement during the computation process. For angular discretization, we utilize high-order discretization near the incident direction, while low-order discretization is applied to other directions. We compare its performance with the classical Monte Carlo simulations for both mono-energetic and multi-energetic photon beams in various media. Numerical results demonstrate the efficiency and accuracy of the algorithm.

This paper introduces well-balanced path-conservative discontinuous Galerkin (DG) methods for two-layer shallow water equations, ensuring exactness for both still water and moving water equilibrium steady states. The approach involves approximating the equilibrium variables within the DG piecewise polynomial space, while expressing the DG scheme in the form of path-conservative schemes. To robustly handle the nonconservative products governing momentum exchange between the layers, we incorporate the theory of Dal Maso, LeFloch, and Murat (DLM) within the DG method. Additionally, linear segment paths connecting the equilibrium functions are chosen to guarantee the well-balanced property of the resulting scheme. The simple “lake-at-rest" steady state is naturally satisfied without any modification, while a specialized treatment of the numerical flux is crucial for preserving the moving water steady state. Extensive numerical examples in one and two dimensions validate the exact equilibrium preservation of the steady state solutions and demonstrate its high-order accuracy. The performance of the method and high-resolution results further underscore its potential as a robust approach for nonconservative hyperbolic balance laws.