arXiv subjects

Guosheng Fu

Publications and source records attributed to Guosheng Fu.

At least 19 recordsLinked to original sources

arXiv2026

Structure-preserving operator splitting for 2.5D ideal MHD with an entropy-stable DGSEM and exactly divergence-free compatible finite elements

We develop a high-order, fully explicit operator-splitting method for 2.5D ideal magnetohydrodynamics on Cartesian meshes. The hydrodynamic subflow is advanced by an entropy-stable discontinuous Galerkin spectral element method equipped with stagewise oscillation elimination and a positivity-preserving limiter. The magnetic--velocity subflow uses compatible finite elements and mass-lumped reconstructions of electric field and current density. Its discrete-curl update exactly preserves the global $H(\mathrm{div})$ divergence-free subspace, while the ideal semidiscretization balances kinetic, magnetic, and internal energy. Curl-form artificial resistivity and a direction-resolved velocity filter return removed magnetic and kinetic energy to internal energy; consequently, the stabilized magnetic stage preserves positive internal energy and satisfies a discrete entropy inequality. The two solvers are composed by a second-order hydrodynamic--magnetic--hydrodynamic Strang splitting, yielding a matrix-free scheme that retains global mass, nodal admissibility, and magnetic divergence on accepted steps. Smooth Alfv\'en-wave and advected-vortex tests reveal an even--odd convergence pattern in the magnetic polynomial degree, verify second-order temporal accuracy, and show that stabilization preserves high-order accuracy. Field-loop, Orszag--Tang, rotor, MHD blast-wave, and Kelvin--Helmholtz calculations demonstrate robust performance for nonsmooth multidimensional flows and show that artificial resistivity suppresses grid-scale magnetic oscillations while retaining the resolved structures.

math.NA↗

arXiv2026

Entropy-Stable and Physical-Constraint-Preserving DGSEM for Symmetry-Reduced General-Relativistic Hydrodynamics on Stationary Spacetimes

We develop an entropy-stable and physical-constraint-preserving discontinuous Galerkin spectral element method for symmetry-reduced general-relativistic hydrodynamics on prescribed stationary spacetimes. Using a local orthonormal transformation, the fluid variables are expressed in a form for which the relativistic hydrodynamic algebra and the admissible set are independent of the spatial metric, while the spacetime geometry enters through stationary coefficients. This separation allows entropy-conservative special-relativistic fluxes to be combined with a compatible discretization of the geometric source terms. On affine tensor-product meshes, the resulting DGSEM is conservative and satisfies a semidiscrete entropy inequality, while the transformed variables provide a convex framework for physical-constraint preservation. For practical stabilization, we use a geometry-only causal speed that is sufficient for both classical local Lax--Friedrichs entropy dissipation and the physical-constraint-preserving Lax--Friedrichs splitting. The fully discrete method combines this stabilization with SSP Runge--Kutta time stepping, oscillation elimination, and conservative local-orthonormal-state scaling. Numerical experiments cover smooth and strongly shocked special-relativistic flows, an axisymmetric jet, stationary Michel accretion, Schwarzschild Bondi--Hoyle flow, and four Kerr accretion cases. The results demonstrate the designed high-order accuracy in smooth regimes and robust performance for demanding relativistic flows on curved stationary backgrounds.

math.NA↗

arXiv2026

Phase-Field Models, Sharp Interface Limits, and Numerical Schemes for Contact Line Dynamics

We study phase-field and sharp-interface models for contact line dynamics of a liquid droplet on a solid substrate within a unified variational framework. The motion of the contact line, where liquid, gas, and solid phases meet, poses a fundamental difficulty in continuum modeling due to the classical stress singularity of no-slip hydrodynamics. Phase-field models regularize this singularity by introducing a thin transition layer of thickness and encoding interfacial effects through a Ginzburg-Landau free energy augmented by a wall energy on the substrate. Starting from the total free energy $E = E_b + E_w$, we analyze two phase-field models: the Allen-Cahn equation and the Cahn-Hilliard equation. Using matched asymptotic expansions as $\delta \to 0$, we recover their corresponding sharp interface limits. In the Allen-Cahn case, the limit yields motion by mean curvature with a contact line law driven by deviations of the dynamic contact angle from Young's angle. In the Cahn-Hilliard case, the limit leads to a Mullins-Sekerka problem with the same form of contact line dynamics. A central result of this work is the identification of consistent gradient-flow structures across both models. The Allen-Cahn dynamics correspond to an $L^2$-gradient flow, while the Cahn-Hilliard dynamics correspond to an $H^{-1}$-gradient flow, and both converge to sharp-interface evolutions that preserve the same energy-dissipation structure. This provides a unified interpretation of contact line motion as a consequence of a single variational principle. Finally, we develop energy-stable numerical schemes based on the minimizing movement principle and establish discrete energy dissipation and well-posedness of the fully discrete problem. Numerical examples confirm that both schemes relax toward the same stationary sharp interface solution while their dynamics reflect the different dissipation mechanisms.

math.AP↗

arXiv2026

Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement

We extend the entropy-stable oscillation-eliminating discontinuous Galerkin spectral element method (ES-OEDG) on curvilinear meshes to adaptive mesh refinement (AMR) grids with nonconforming interfaces. The formulation targets two-dimensional curvilinear quadrilateral meshes under a 2:1 refinement constraint, allowing a single level of hanging nodes. Elementwise volume discretization and geometric mapping are retained, while oscillation elimination and interface coupling are adapted for nonconforming interfaces. A central contribution is the design and analysis of numerical fluxes for such interfaces. We construct an entropy-stable flux that ensures global conservation and a semi-discrete entropy inequality. However, for polynomial degree N >= 2, negative entries in nonconforming interpolation operators lead to loss of formal high-order consistency. To address this, we propose a mortar-based flux that preserves high-order accuracy by interpolating at the solution level and evaluating standard two-point fluxes on fine-side mortars, at the cost of losing provable entropy stability. We also extend the Zhang--Shu positivity-preserving framework to curvilinear AMR meshes. Under forward Euler time stepping and a suitable CFL condition, the scheme using either flux preserves positivity of cell-average density and pressure. Combined with the Zhang--Shu limiter, this yields a fully discrete scheme maintaining admissibility at all nodal points. We further incorporate shock-indicator-based AMR and a conservative, positivity-preserving data transfer procedure between successive meshes, resulting in a robust and efficient algorithm. Numerical experiments on Cartesian and curvilinear AMR grids confirm high-order accuracy and robustness.

math.NA↗

arXiv2026

From molecular dynamics to kinetic models: data-driven generalized collision operators in 1D3V plasmas

We present a data-driven approach for constructing generalized collisional kinetic models for inhomogeneous plasmas in one-dimensional physical space and three-dimensional velocity space (1D-3V). The collision operator is directly learned from micro-scale molecular dynamics (MD) and accurately accounts for the unresolved particle interactions over a broad range of plasma conditions. Unlike the standard Landau operator, the present operator takes an anisotropic, non-stationary form that captures the heterogeneous collisional energy transfer arising from the many-body interactions, which is crucial for plasma kinetics beyond the weakly coupled regime. Efficient numerical evaluation is achieved through a low-rank tensor representation with $O(N \log N)$ computational complexity. The constructed kinetic equation strictly preserves conservation laws and physical constraints and therefore, enables us to develop an explicit second-order, energy-conserving scheme that ensures fully discrete conservation of mass and total energy. Numerical results demonstrate that the present model accurately predicts both transport coefficients and several 1D-3V kinetic processes compared with MD simulations across a broad range of densities and temperatures in spatially inhomogeneous settings. This work provides a systematic pathway for bridging micro-scale MD and inhomogeneous plasma kinetic descriptions where empirical models show limitation.

physics.plasm-ph↗

arXiv2026

An entropy-stable oscillation-eliminating dgsem for the euler equations on curvilinear meshes

We develop an entropy-stable high-order numerical method for the two-dimensional compressible Euler equations on general curvilinear meshes. The proposed approach is based on a nodal discontinuous Galerkin spectral element method (DGSEM) that satisfies the summation-by-parts (SBP) property. At the semidiscrete level, entropy stability is established through the SBP structure and the discrete metric identities associated with curvilinear coordinate mappings. By incorporating entropy-stable numerical fluxes at element interfaces, a global discrete entropy inequality is obtained. To further control nonphysical oscillations near strong discontinuities, the entropy-stable DG formulation is combined with a modified oscillation-eliminating discontinuous Galerkin (OEDG) method, which was originally proposed in [59]. We observe that the zero-order damping coefficient in the original OEDG method naturally serves as an effective shock indicator, which enables localization of the oscillation control mechanism and significantly reduces computational cost. Moreover, while the original OEDG formulation relies on local orthogonal modal bases and is primarily restricted to simplicial meshes, we reformulate the OE procedure using projection operators, allowing for a systematic extension to general curvilinear meshes. The resulting method preserves conservation and entropy stability while effectively suppressing spurious oscillations. A series of challenging numerical experiments is presented to demonstrate the accuracy, robustness, and effectiveness of the proposed entropy-stable OEDG method on both Cartesian and curvilinear meshes.

math.NA↗

arXiv2026

The proximal Galerkin method for non-symmetric variational inequalities

We introduce the proximal Galerkin (PG) method for non-symmetric variational inequalities. The proposed approach is asymptotically mesh-independent and yields constraint-preserving approximations. We present both a conforming PG formulation and a hybrid mixed first-order system variant (FOSPG). We establish optimal a priori error estimates for each variant, which are verified numerically. We conclude by applying the method to American option pricing, free boundary problems in porous media, advection-diffusion with a semipermeable boundary, and the enforcement of discrete maximum principles.

math.NA↗

arXiv2026

A Schr\"odinger-Based Dispersive Regularization Approach for Numerical Simulation of One-Dimensional Shallow Water Equations

We propose a novel dispersive regularization framework for the numerical simulation of the one-dimensional shallow water equations (SWE). The classical hyperbolic system is regularized by a third-order dispersive term in the momentum equation, which renders the system equivalent, via the Madelung transform, to a defocusing cubic nonlinear Schr\"odinger equation with a drift term induced by bottom topography. Instead of solving the shallow water equations directly, we solve the associated Schr\"odinger equation and recover the hydrodynamic variables through a simple postprocessing procedure. This approach transforms the original nonlinear hyperbolic system into a semilinear complex-valued equation, which can be efficiently approximated using a Strang time-splitting method combined with a spectral element discretization in space. Numerical experiments demonstrate that, in subcritical regimes without shock formation, the Schr\"odinger regularization provides an $O(\varepsilon)$ approximation to the classical shallow water solution, where $\varepsilon$ denotes the regularization parameter. Importantly, we observe that this convergence behavior persists even in the presence of moving wetting--drying interfaces, where vacuum states emerge and standard shallow water solvers often encounter difficulties. These results suggest that the Schr\"odinger-based formulation offers a robust and promising alternative framework for the numerical simulation of shallow water flows with dry states.

math.NA↗

arXiv2025

Finite Element Representation Network (FERN) for Operator Learning with a Localized Trainable Basis

We propose a finite-element local basis-based operator learning framework for solving partial differential equations (PDEs). Operator learning aims to approximate mappings from input functions to output functions, where the latter are typically represented using basis functions. While non-learnable bases reduce training costs, learnable bases offer greater flexibility but often require deep network architectures with a large number of trainable parameters. Existing approaches typically rely on deep global bases; however, many PDE solutions exhibit local behaviors such as shocks, sharp gradients, etc., and in parametrized PDE settings, these localized features may appear in different regions of the domain across different training and testing samples. Motivated by the use of local bases in finite element methods (FEM) for function approximation, we develop a shallow neural network architecture that constructs adaptive FEM bases. By adopting suitable activation functions, such as ReLU, the FEM bases can be assembled exactly within the network, introducing no additional approximation error in the basis construction process. This design enables the learning procedure to naturally mimic the adaptive refinement mechanism of FEM, allowing the network to discover basis functions tailored to intrinsic solution features such as shocks. The proposed learnable adaptive bases are then employed to represent the solution (output function) of the PDE. This framework reduces the number of trainable parameters while maintaining high approximation accuracy, effectively combining the adaptivity of FEM with the expressive power of operator learning. To evaluate performance, we validate the proposed method on seven families of PDEs with diverse characteristics, demonstrating its accuracy, efficiency, and robustness.

math.NA↗

arXiv2025

A four-field mixed formulation for incompressible finite elasticity

In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [Gopalakrishnan J., Lederer P., and Sch\"oberl J., A mass conserving mixed stress formulation for the Stokes equations, IMA Journal of Numerical Analysis 40(3), 1838-1874 (2019)], involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu-Washizu principle, introducing the displacement gradient and 1st Piola-Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical experiments.

math.NA↗

arXiv2024

A locally-conservative proximal Galerkin method for pointwise bound constraints

We introduce the first-order system proximal Galerkin (FOSPG) method, a locally mass-conserving, hybridizable finite element method for solving heterogeneous anisotropic diffusion and obstacle problems. Like other proximal Galerkin methods, FOSPG finds solutions by solving a recursive sequence of smooth, discretized, nonlinear subproblems. We establish the well-posedness and convergence of these nonlinear subproblems along with stability and error estimates under low regularity assumptions for the linearized equations obtained by solving each subproblem using Newton's method. The FOSPG method exhibits several advantages, including high-order accuracy, discrete maximum principle or bound-preserving discrete solutions, and local mass conservation. It also achieves prescribed solution accuracy within asymptotically mesh-independent numbers of subproblems and linear solves per subproblem iteration. Numerical experiments on benchmarks for anisotropic diffusion and obstacle problems confirm these attributes. Furthermore, an open-source implementation of the method is provided to facilitate broader adoption and reproducibility.

math.NA↗

arXiv2024

Efficient Computation of Mean field Control based Barycenters from Reaction-Diffusion Systems

We develop a class of barycenter problems based on mean field control problems in three dimensions with associated reactive-diffusion systems of unnormalized multi-species densities. This problem is the generalization of the Wasserstein barycenter problem for single probability density functions. The primary objective is to present a comprehensive framework for efficiently computing the proposed variational problem: generalized Benamou-Brenier formulas with multiple input density vectors as boundary conditions. Our approach involves the utilization of high-order finite element discretizations of the spacetime domain to achieve improved accuracy. The discrete optimization problem is then solved using the primal-dual hybrid gradient (PDHG) algorithm, a first-order optimization method for effectively addressing a wide range of constrained optimization problems. The efficacy and robustness of our proposed framework are illustrated through several numerical examples in three dimensions, such as the computation of the barycenter of multi-density systems consisting of Gaussian distributions and reactive-diffusive multi-density systems involving 3D voxel densities. Additional examples highlighting computations on 2D embedded surfaces are also provided.

math.OC↗

arXiv2024

Mean field control of droplet dynamics with high order finite element computations

Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formation such as droplet merging, splitting, and transport. This paper studies a class of mean field control formulations for these droplet dynamics, which can be used to control and manipulate droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. As an example, a lubrication equation for a thin volatile liquid film laden with an active suspension is developed, with control achieved through its activity field. Lastly, we apply the primal-dual hybrid gradient algorithm with high-order finite element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism.

math.OC↗

arXiv2023

Generalized optimal transport and mean field control problems for reaction-diffusion systems with high-order finite element computation

We design and compute a class of optimal control problems for reaction-diffusion systems. They form mean field control problems related to multi-density reaction-diffusion systems. To solve proposed optimal control problems numerically, we first apply high-order finite element methods to discretize the space-time domain and then solve the optimal control problem using augmented Lagrangian methods (ALG2). Numerical examples, including generalized optimal transport and mean field control problems between Gaussian distributions and image densities, demonstrate the effectiveness of the proposed modeling and computational methods for mean field control problems involving reaction-diffusion equations/systems.

math.OC↗

arXiv2023

Two Finite Element Approaches For The Porous Medium Equation That Are Positivity Preserving And Energy Stable

In this work, we present the construction of two distinct finite element approaches to solve the Porous Medium Equation (PME). In the first approach, we transform the PME to a log-density variable formulation and construct a continuous Galerkin method. In the second approach, we introduce additional potential and velocity variables to rewrite the PME into a system of equations, for which we construct a mixed finite element method. Both approaches are first-order accurate, mass conserving, and proved to be unconditionally energy stable for their respective energies. The mixed approach is shown to preserve positivity under a CFL condition, while a much stronger property of unconditional bound preservation is proved for the log-density approach. A novel feature of our schemes is that they can handle compactly supported initial data without the need for any perturbation techniques. Furthermore, the log-density method can handle unstructured grids in any number of dimensions, while the mixed method can handle unstructured grids in two dimensions. We present results from several numerical experiments to demonstrate these properties.

math.NA↗

arXiv2023

High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems

We design and compute first-order implicit-in-time variational schemes with high-order spatial discretization for initial value gradient flows in generalized optimal transport metric spaces. We first review some examples of gradient flows in generalized optimal transport spaces from the Onsager principle. We then use a one-step time relaxation optimization problem for time-implicit schemes, namely generalized Jordan-Kinderlehrer-Otto schemes. Their minimizing systems satisfy implicit-in-time schemes for initial value gradient flows with first-order time accuracy. We adopt the first-order optimization scheme ALG2 (Augmented Lagrangian method) and high-order finite element methods in spatial discretization to compute the one-step optimization problem. This allows us to derive the implicit-in-time update of initial value gradient flows iteratively. We remark that the iteration in ALG2 has a simple-to-implement point-wise update based on optimal transport and Onsager's activation functions. The proposed method is unconditionally stable for convex cases. Numerical examples are presented to demonstrate the effectiveness of the methods in two-dimensional PDEs, including Wasserstein gradient flows, Fisher--Kolmogorov-Petrovskii-Piskunov equation, and two and four species reversible reaction-diffusion systems.

math.NA↗

arXiv2023

$hp$-Multigrid preconditioner for a divergence-conforming HDG scheme for the incompressible flow problems

In this study, we present an $hp$-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier-Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The $hp$-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we developed a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix-Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput. Methods Appl. Mech. Engrg., 311 (2016)), which allows for the direct application of geometric multigrid theory on the Crouzeix-Raviart discretization. The numerical experiments demonstrate the robustness of the proposed $hp$-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi & Olshanskii (SIAM J. Sci. Comput., 28(6) (2006)) and Farrell et al. (SIAM J. Sci. Comput., 41(5) (2019)), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier-Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to $10^3$.

math.NA↗

arXiv2023

High-order variational Lagrangian schemes for compressible fluids

We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit time stepping is used for the temporal discretization, which allows for a much larger time step size for stability compared to explicit methods, especially for low-Mach number flows and/or on highly distorted meshes. Ample numerical results are presented to showcase the good performance of our proposed scheme.

math.NA↗