Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard (FCH) energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the “morphological complexity” observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split directly into networks of interfaces. We evaluate a number of schemes that use second order backward differentiation formula (BDF2) type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a preconditioned steepest descent (PSD) nonlinear solver or upon linearly implicit time discretization based on the standard implicit-explicit (IMEX) and the scalar auxiliary variable (SAV) approaches. We add an exponential time differencing (ETD) scheme for comparison purposes. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper “preconditioning” to achieve robust performance that can enhance efficiency by several orders of magnitude. The nonlinear PSD scheme achieves the smallest global discretization error at fixed local truncation error, however the IMEX and SAV schemes are the most computationally efficient as measured by the number of Fast Fourier Transform (FFT) calls required to achieve a desired global error. Indeed the performance of the SAV scheme directly mirrors that of IMEX, modulo a factor of 1.4 in FFT calls for the auxiliary variable system.

In this paper, the unconditionally energy-stable and orthonormality-preserving iterative scheme proposed in [X. Wang et al. (2024), J. Comput. Phys., 498:112670] is extended both theoretically and numerically, including (i) the exchange-correlation energy is introduced into the model for a more comprehensive description of the quantum system, utilizing the local density approximation used by the National Institution of Science and Technology Standard Reference Database; (ii) both the unconditional energy-stability and orthonormality-preservation are attained in the newly derived scheme; (iii) a $C^0$ tetrahedral spectral element method is adopted for the quality spatial discretization, of which a quality initial condition can be designed using low order one for effectively accelerating the simulation. A series of numerical experiments validate the effectiveness of our method, encompassing various atoms and molecules. All the computations successfully reveal the anticipated spectral accuracy and the exponential error dependence to the cubic root of the degree of freedom number. Moreover, the efficiency of the extended framework is discussed in detail on updating schemes.

Recently, physics-informed neural networks (PINNs) have been shown to be a simple and efficient method for solving PDEs empirically. However, the numerical analysis of PINNs is still incomplete, especially why over-parameterized PINNs work remains unknown. This paper presents the first convergence analysis of the over-parameterized PINNs for the Laplace equations with Dirichlet boundary conditions. We demonstrate that the convergence rate can be controlled by the weight norm, regardless of the number of parameters in the network.

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 Schrödinger 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 $L^2$-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 extend the weighted discontinuous Galerkin finite element method (WDG) on polygonal grids for solving the dual-porosity-Navier-Stokes model. The Navier-Stokes model describes the free flow in conduits, while the dual-porosity model describes the fluid flow in a medium composed of matrix and microfractures. These two models are coupled through four physically meaningful interface conditions. We obtain the existence and local uniqueness of the solution, as well as the optimal error estimate, under appropriate small data conditions that maintain physical properties. Through numerical experiments, the advantages of the numerical method are verified, such as the optimal convergence rate of the numerical solution to different mesh types and numerical schemes, the performance of the classical upwind scheme combined with the Picard iteration method in handling small viscosity problems, the flow around a horizontal production wellbore with open-hole completion, the different application simulation of multistage hydraulic fractured horizontal wellbore with cased hole completion, as well as the simulation of fluid flow characteristics around macro-fractures.

In this article, we investigate some issues related to the quantification of uncertainties associated with the electrical properties of graphene nanoribbons. The approach is suited to understand the effects of missing information linked to the difficulty of fixing some material parameters, such as the band gap, and the strength of the applied electric field. In particular, we focus on the extension of particle Galerkin methods for kinetic equations in the case of the semiclassical Boltzmann equation for charge transport in graphene nanoribbons with uncertainties. To this end, we develop an efficient particle scheme which allows us to parallelize the computation and then, after a suitable generalization of the scheme to the case of random inputs, we present a Galerkin reformulation of the particle dynamics, obtained by means of a generalized Polynomial Chaos approach, which allows the reconstruction of the kinetic distribution. As a consequence, the proposed particle-based scheme preserves the physical properties and the positivity of the distribution function also in the presence of a complex scattering in the transport equation of electrons. The impact of the uncertainty of the band gap and applied field on the electrical current is analysed.

Mixed grids offer advantages in grid generation, calculation efficiency and numerical accuracy. The $p$-weighted limiter works effectively on 1D, triangular and tetrahedral grids for the discontinuous Galerkin (DG) method. Whereas the generalization to unstructured mixed grids for nodal DG methods is not straightforward. On quadrilateral grids, the tensor-product of solution points for nodal DG methods makes it complex and time-consuming to construct bi-linear candidate polynomials and perform weighted summation with the tensor-product polynomials. A simple yet effective way is proposed to address this issue. The linear candidate polynomial is constructed by interpolating values on the face and performing a forward difference in the central troubled cell. The truncation strategy on the linear weights is also improved. The weighted summation is carried out dimension-by-dimension on local coordinates as the 1D case. Since the limiting only depends on the central cell type, the final implementation for nodal DG methods on mixed grids is straightforward. A parameter-free shock detector is proposed to identify the troubled cells near shock waves. Several inviscid, laminar and turbulent flows with shocks are employed to showcase the efficiency of the improved $p$-weighted limiter and shock detector on unstructured mixed grids for nodal DG schemes.

Combining robustness and high accuracy is one of the primary challenges in the magnetohydrodynamics (MHD) field of numerical methods. This paper investigates two critical physical constraints: wave order and positivity-preserving (PP) properties of the high-resolution HLLD Riemann solver, which ensures the positivity of density, pressure, and internal energy. This method’s distinctiveness lies in its ability to ensure that the wave characteristic speeds of the HLLD Riemann solver are strictly ordered. A provably PP HLLD Riemann solver based on the Lagrangian setting is established, which can be viewed as an extension of the PP Lagrangian method in hydrodynamics but with more and stronger constraint condition. In addition, the above two properties are ensured on moving grid method by employing the Lagrange-to-Euler transform. Meanwhile, a novel multi-moment constrained finite volume method is introduced to acquire third order accuracy, and practical limiters are applied to avoid numerical oscillations. Selected numerical benchmarks demonstrate the robustness and accuracy of our methods.

We present two new algebraic multilevel hierarchical matrix algorithms to perform fast matrix-vector product (MVP) for N-body problems in d dimensions, namely efficient $\mathcal{H}^2_∗$ (fully nested algorithm, i.e., $\mathcal{H}^2$ matrix algorithm) and $(\mathcal{H}^2+\mathcal{H})_∗$ (semi-nested algorithm, i.e., cross of $\mathcal{H}^2$ and $\mathcal{H}$ matrix algorithms). The efficient $\mathcal{H}^2_*$ and $(\mathcal{H}^2+\mathcal{H})_∗$ hierarchical representations are based on our recently introduced weak admissibility condition in higher dimensions (Khan et al., J. Comput. Phys. 2024), where the admissible clusters are the far-field and the vertex-sharing clusters. Due to the use of nested form of the bases, the proposed hierarchical matrix algorithms are more efficient than the non-nested algorithms ($\mathcal{H}$ matrix algorithms). We rely on purely algebraic low-rank approximation techniques (e.g., ACA (Bebendorf et al., Computing 2003) and NCA (Bebendorf et al., Numer. Math. 2012; Gujjula and Ambikasaran, arXiv:2203.14832 2022; Zhao et al., IEEE Trans. Microw. Theory Tech. 2019)) and develop both algorithms in a black-box (kernel-independent) fashion. The initialization time of the proposed algorithms scales quasi-linearly, i.e., complexity $\mathcal{O}(N{\rm log}^α (N)),$ $α ≥0$ and small. Using the proposed hierarchical representations, one can perform the MVP that scales at most quasi-linearly. Another noteworthy contribution of this article is that we perform a comparative study of the proposed algorithms with different algebraic (NCA or ACA-based compression) fast MVP algorithms (e.g.,$\mathcal{H}^2, $ $\mathcal{H},$ etc.) in 2D and 3D ($d = 2,3$). The fast algorithms are tested on various kernel matrices and applied to get fast iterative solutions of a dense linear system arising from the discretized integral equations and radial basis function interpolation. The article also discusses the scalability of the algorithms and provides various benchmarks. Notably, all the algorithms are developed in a similar fashion in C++ and tested within the same environment, allowing for meaningful comparisons. The numerical results demonstrate that the proposed algorithms are competitive to the NCA-based standard $\mathcal{H}^2$ matrix algorithm (where the admissible clusters are the far-field clusters) with respect to the memory and time. The C++ implementation of the proposed algorithms is available at https://github.com/riteshkhan/H2weak/.