ABSTRACT Heterogeneous numerical models (HNMs) combine conventional discretization modules such as finite elements with nonconventional data‐driven and reduced‐order modules. HNMs can improve computational efficiency and enable simulations of multi‐physics systems in which one or more constituent components lack first‐principles descriptions and must be solved by a data‐driven method. A natural choice for solving HNMs is partitioned solvers that are minimally intrusive and only require exchange of interface information between the modules. In this article, we formulate and demonstrate a partitioned solution of a coupled HNM for a model transmission problem in which each subdomain can be solved by equation‐free and/or equation‐based modules. Specifically we consider HNMs combining Dynamic Mode Decomposition (DMD) [1] modules with other DMD modules or conventional Finite Elements. The subdomains are decoupled by specifying the interface flux exchanged between them as a Neumann boundary condition for the modules. At each time step, this flux is estimated by a dynamic flux surrogate that is learned offline from a combination of flux and state training data. We test the partitioned HNM approach using single‐ and multi‐material configurations of the model problem. Numerical results show that the dynamic flux surrogate approach enables accurate, stable, and efficient solution of the coupled problem.
ABSTRACT In this work, we present an extension of the semi‐discrete Lagrangian‐Eulerian numerical scheme for diffusive‐dispersive conservation law problems, including a jump discontinuous flux function. As in the one‐dimensional scalar hyperbolic case, the no‐flow curves also organize the geometry of the method. On the computational side, we successfully reproduced one‐dimensional scalar problems from the literature, and to validate robustness under more extreme and challenging conditions, we extended the experiments to problems that go beyond what is currently available. In the two‐dimensional case, still in an exploratory manner, we obtained consistent results that indicate a natural extension of the method to multidimensional problems, guided by the no‐flow curves. On the theoretical side, with the coefficients of the diffusive and dispersive terms in balance, we showed the convergence of the numerical approximations to weak solutions and, in the hyperbolic limit problem, the convergence to the entropy solution. For convergence to the weak solution, we employed the Compensated Compactness Theorem, while for entropy convergence we used a Kružkov‐type criterion suitable for jump discontinuities in the flux function. Together, the results in this work indicate that the method is consistent, robust, and naturally extensible to more general configurations.
ABSTRACT Deep‐learning based partial differential equation (PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into highly nonlinear optimization problems of neural network parameters. This work does a survey and comparative study of several deep‐learning solvers proposed a few years back, including PINN, WAN, DRM, and VPINN. Numerical results are provided to make comparisons among them and address the importance of loss formulation and the optimization method. A rigorous error analysis for PINN is presented, and brief error estimate results for other methods are discussed. Finally, we discuss the current limitations and bottlenecks of these methods.
ABSTRACT In this paper, we present a general operational matrix of integration for the extended Jacobi wavelet. Furthermore, we apply this framework to obtain an approximate solution of the nonlinear Rosenau–Hyman equation. The proposed method is based on a wavelet technique that leverages the properties of Jacobi polynomials to enhance computational accuracy. We perform a comparative analysis of the approximate solution for various parameter values to investigate its behavior and precision. In addition, we compute the absolute error, relative error, as well as the and errors for different parameter settings, thereby demonstrating the efficiency and reliability of the proposed method. Moreover, we prove some theorems regarding the convergence and error estimates of the method with respect to different norms.
ABSTRACT In this report, we present a family of regularized ensemble methods for fast simulation of the Navier‐Stokes flow ensembles. The ensemble algorithms are constructed so that all ensemble members share the same coefficient matrix after spatial discretization to facilitate the use of efficient direct or iterative linear solvers for solving the corresponding linear systems. Although this idea has been tested in many papers in recent years and shown to be very effective in reducing the computational cost and storage for computing a large ensemble of flow data, an outstanding issue for computing nonlinear flows is that the fluctuation‐induced timestep condition deteriorates fast as the Reynolds number increases, and the time step needs to be taken small enough for the method to stay stable. To help relax this timestep condition, we propose a family of regularized ensemble methods that are second order in time, featuring a regularization term that is proportional to the discrete curvature of the solution. We prove the long‐time stability and convergence of the methods under a timestep condition. Extensive numerical tests are presented to confirm theoretical results and demonstrate the advantages of using this curvature regularization that mitigates the severity of the timestep condition.
ABSTRACT In this paper, we develop and analyze a high‐order numerical scheme for a two‐dimensional Caputo time‐fractional Black–Scholes (TFBS) equation whose solution exhibits a weak singularity at the initial time . To effectively handle this singular behavior, a fast L2‐ discretization on a graded mesh is employed in the temporal direction, while the Galerkin finite element method (FEM) is used for the spatial discretization. A rigorous stability and convergence analysis of the proposed scheme is performed. The theoretical analysis establishes that the scheme achieves second‐order accuracy in the spatial direction and a convergence rate of order in the temporal direction. The proposed method is validated through a test problem with a known analytical solution. Moreover, its practical applicability is further demonstrated by solving two real‐world financial problems involving the pricing of European call and put options. The numerical results confirm that the graded mesh yields significantly more accurate solutions compared to the uniform mesh due to its ability to capture the weak singularity present in the solution at the initial time.
ABSTRACT In this paper, we investigate the initial value problem for a nonlinear space‐time‐fractional diffusion equation with a Caputo time‐fractional derivative of order , a fractional Laplacian of order , and a nonlinear source term . Since the problem is ill‐posed, we propose a spectral cut‐off regularization method to solve the problem in a stable way. The regularized problem is proved to be well‐posed. Then, we obtain some error estimates between the exact solution and the regularized solution. We also propose a numerical algorithm to solve the regularization problem numerically. The numerical algorithm is proved to converge to the regularized solution with an error of , where is the time step, is the number of Fourier modes, and . Numerical examples in one and two dimensional spaces confirm our theoretical results of convergent rates, show the robustness of the algorithm under significant noise levels.
ABSTRACT The mixed population balance equation including aggregation, collision‐induced breakage, and growth has significant usage in particulate modeling. These models are highly nonlinear and difficult to solve accurately. Discretization‐based sectional methods often lead to complex, grid‐dependent formulations. This reduces computational efficiency and may affect numerical stability. Extending these approaches to multivariate models is also challenging. To overcome these issues, a semi‐analytical mesh‐free method based on a homotopy approach is proposed. The method is further developed to handle multivariate models in a systematic way. Temimi‐Ansari method [Comput. Math. Appl., 2011, 61(2), 203‐210] is also applied to the same problem to examine the performance of the proposed approach. The accuracy is evaluated through the prediction of the solution and the associated moment functions. Numerical examples based on equi‐partition of kinetic energy (EKE) kernels, relevant to droplet behavior, are considered. The results show that the proposed method is simple, efficient, and provides accurate solutions.
ABSTRACT This paper proposes a family of fully discrete, second‐order linearly extrapolated curvature‐stabilized ‐schemes for solving the diffusive Peterlin viscoelastic model. By varying , we obtain distinct second‐order temporal schemes: at , the algorithm reduces to the BDF2 extrapolation method, while yields the Crank‐Nicolson extrapolation. We rigorously establish the existence and uniqueness of the discrete solutions, along with optimal error estimates for both the velocity and the conformation tensor under suitable regularity assumptions. The analysis highlights the role of the stabilization in ensuring stability without compromising second‐order accuracy. Three numerical experiments validate the theoretical analysis, demonstrating the scheme's second‐order convergence, and effectiveness in simulating the diffusive Peterlin model.
ABSTRACT The Ginzburg‐Landau‐Schrödinger equation (GLSE) can accurately capture the phase transition and behaviors in nonlinear dynamics. It is challenging to develop a numerical scheme that preserves the energy dissipation property while maintaining high computational efficiency. This inspires us to propose a fully discrete scheme that is linear, decoupled, and unconditionally energy‐stable in this article. The time discretization adopts the relaxation method of regularized energy reconstruction (RRER), combined with the Crank‐Nicolson scheme on a staggered time mesh. The spatial discretization employs the symmetric interior penalty discontinuous Galerkin (SIPG) method based on least‐squares fitting (LSF) and patch reconstruction. The fully discrete scheme can achieve second‐order accuracy in time and arbitrary‐order accuracy in the DG‐norm with only one degree of freedom per element. The unconditional energy stability is rigorously proved, and it is demonstrated that the modified energy is a second‐order temporal approximation of the original energy. Numerical examples validate the effectiveness of the algorithm.
ABSTRACT In this paper, we propose a deep learning method called the three‐stage physics‐informed neural network (TS‐PINN) for solving inverse problems of piecewise‐continuous variable coefficients of partial differential equations (PDEs). Unlike the existing PINN‐type methods that typically use a composite loss functional to jointly optimize the solution and the coefficient, the novel TS‐PINN method employs two different neural networks: the solution and coefficient networks, and decomposes the complex training process into three stages according to all the information at hand. The solution network learns an initial approximation of the PDE solution in the first stage. Based on this approximation, the coefficient network estimates the unknown coefficients in the second stage. With the two networks learned in the first two stages, in the third stage, the two networks are trained together on newly constructed training sets. Our numerical tests show that the TS‐PINN is effective and accurate for various types of variable coefficients, including polynomial, trigonometric, exponential, space‐time dependent, and piecewise‐continuous functions. We also provide an in‐depth analysis of the TS‐PINN regarding the necessity of the three‐stage training and the anti‐noise analysis.
ABSTRACT This paper presents a simple linear, maximum bound principle‐preserving L1 scheme for the time‐fractional Allen‐Cahn (TFAC) equation. The scheme is built upon a novel extrapolation‐projection strategy that linearizes the nonlinear term and leads to a linear system with constant coefficients at each time step, substantially reducing computational cost and providing strong long‐time robustness. Although the extrapolation‐projection strategy does not alter the asymptotic convergence order inherent to the L1 approximation, the results indicate improved long‐time behavior through a more effective treatment of the nonlinear term. Moreover, the simulations exhibit a robust monotonic decay of a discrete energy functional. Comparative results show a clear improvement in numerical resolution over standard backward linearization approaches. Furthermore, the extrapolation‐projection framework is sufficiently general and can be readily extended to generalized TFAC equations with nonlinear free energy densities, including higher‐order polynomial and logarithmic potentials.
We propose a linear variable-step exponential time-differencing method for the incompressible Navier–Stokes equations in vorticity–streamfunction formulation on a two-dimensional periodic box. The method consists of a second-order scheme and an embedded first-order variant, yielding a natural mechanism for adaptive time stepping and a posteriori error control. Each time step requires only uniquely solvable linear problems: two heat equation solves, efficiently handled by Fourier methods in the periodic setting, and one linear scalar auxiliary-variable equation, evaluated via Laplace transform and Talbot's numerical inverse transform. The construction combines the ETD framework, a mean-reverting scalar auxiliary variable (mr-SAV), and second-order extrapolation of the nonlinear term. The mean-reverting correction enables long-time stability while preserving full linearity, distinguishing the method from related mr-SAV schemes that require nonlinear algebraic solves. We prove unconditional long-time stability: for uniformly bounded L^2 forcing, the discrete vorticity remains bounded in L^∞(0,∞;L^2) for all Reynolds numbers and time-step sizes. Numerical experiments
ABSTRACT Existing numerical studies for the strongly nonlinear Benjamin–Bona–Mahony–Burgers (BBM‐Burgers) equation focus on high‐order discrete schemes, but these often suffer from trade‐offs among accuracy, stability, and computational efficiency. This article presents a linear compact difference scheme for the BBM‐Burgers periodic initial‐boundary value problem, achieving temporal second‐order and spatial fourth‐order accuracy. We introduce an auxiliary variable to mitigate the equation's high‐order complexity, thereby avoiding cumbersome inner‐product manipulations and streamlining the scheme. A quartic approximation is applied to the nonlinear term to preserve high precision. Via the energy method and mathematical induction, we rigorously establish the scheme's conservation property, ‐norm convergence, and stability. Numerical experiments validate the theoretical convergence order and accurately resolve the intricate dynamical behavior of BBM‐Burgers solutions.
ABSTRACT While recent Anderson acceleration (AA) convergence theory [Pollock et al., IMA Num. An. , 2021] requires that the AA optimization norm match the Hilbert space norm associated with the fixed point operator, in implementations the norm is the most common choice. So far there is little research done regarding this discrepancy. To address this issue, we consider AA applied to the Picard iteration for the Navier–Stokes equations (NSE) with varying choices of the AA optimization norm. We prove a sharpened and generalized convergence estimate with the AA optimization norm, and an essentially equivalent (in terms of associated Lipschitz constants) result for when is used. While no analogous theory seems possible for , numerical tests were run to compare varying choices of AA optimization norms. These tests revealed similar convergence for and in terms of iterations needed for convergence and usually but not always similar for : for flow past a cylinder, convergence using performs significantly worse.
This article proposes an optimal convergence analysis of two model problems: the stationary Maxwell equations, which represent the ‐elliptic problem, and the time‐dependent Maxwell equations in cold plasma. We employ skeletal discontinuous Galerkin (DG) methods for spatial discretization. First, we introduce a skeletal DG method for the ‐elliptic problem with variable coefficients and discuss the optimal convergence analysis in the energy and norms. Next, we propose a continuous in time skeletal DG method for the Maxwell problem in cold plasma. The proof of error converging at an optimal rate for the cold plasma equations in and discrete energy norms hinges on a suitably defined Ritz projection derived from the previously discussed stationary Maxwell problem. We also present numerical computations in two and three dimensions for the stationary and time‐dependent Maxwell equations, including implicit and explicit time integration techniques for the time‐dependent case. These computations verify the theoretical rates we have presented.
ABSTRACT Minimizers of phase‐field models are commonly computed by evolving gradient‐flow dynamics, such as , , or Wasserstein gradient flows, toward equilibrium. However, phase‐field energies often possess complicated energy landscapes, and the corresponding relaxation dynamics may involve long‐time, multi‐stage evolution. In many applications, the main objective is to identify the self‐assembled equilibrium morphology, rather than to resolve the entire dynamical pathway leading to it. In such cases, direct simulation of the underlying gradient flow can be unnecessarily expensive. Motivated by this observation, we introduce a Nesterov accelerated gradient flow framework for phase‐field models of block copolymer systems and develop corresponding Nesterov accelerated gradient descent schemes. We prove that a modified energy dissipation law, held at the continuous level, is inherited by the proposed discrete schemes. We further compare the accelerated scheme with a BDF2 discretization of the gradient flow. Numerical experiments show that the proposed Nesterov accelerated gradient descent scheme reduces the computational cost by a factor of approximately six, while retaining accuracy comparable to classical gradient‐flow solvers.
In this paper, we develop a second-order structure-preserving Crank-Nicolson difference scheme for a Riesz space fractional regularized long wave-Korteweg-de Vries (RLW-KdV) equation. The proposed scheme can preserve mass and energy in a discrete sense. Based on the Brouwer fixed-point theorem, the existence of the difference solution is proved. The convergence, stability, and uniqueness of the difference scheme are proved by using the energy method. It is shown to be convergent of accuracy in the discrete -norm with the space step size and time step size . In addition, the resulting system of nonlinear equations is solved by the Newton method. When the fractional order is two, all those results are in accord with the difference scheme proposed for the classical RLW-KdV equation. Some numerical examples are given to confirm the theoretical results.
We propose a weak Galerkin finite element method for the Westervelt's model of ultrasound waves on polygonal meshes. Specifically, we investigate the spatial discretization of Westervelt's quasi-linear, strongly damped wave equation using high-order weak Galerkin discretization. The primary challenges in the numerical analysis include managing the nonlinear terms in the model and preventing the equation from degenerating. We avoid the degeneracy of the semi-discrete Westervelt's equation by employing inverse estimates and the stability and approximation properties of the L-2 projection. Our convergence analysis relies on the Banach fixed-point theorem, along with a stability and convergence analysis of a linear diffusive viscous wave equation with variable coefficients for the first and second time derivatives. This approach yields optimal convergence rates in L-2-based spatial norms for sufficiently small data and mesh size, given an appropriate choice of initial data. Numerical experiments conducted in two-dimensional settings illustrate the theoretical convergence results.
In this paper we present and analyze a stabilized linear fully discrete scheme for the Cahn-Hilliard equation with dynamic boundary condition proposed by Liu and Wu. We employ a higher order asymptotic expansion (up to second-order temporal accuracy) and construct a correction term (higher-order infinitesimal). This ensures that the discrete error function has zero mean, laying the foundation for its analysis in norm. By combining rough error estimates and refined error estimates, a complete convergence analysis of the scheme is accomplished. Specifically, by treating the numerical solution as a small perturbation of the exact solution, we prove the uniform boundedness of the numerical solution and its temporal difference in the discrete norm based on the results of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, with the help of bound of the numerical solution and its temporal difference. Our stability analysis requires neither the global Lipschitz assumption nor cut-off for the nonlinear term, which significantly improves the corresponding findings in the previous literature. Some numerical experiments are performed to verify the optimal convergence order and the robustness of the proposed scheme.
