In this paper, we further extend the derivative-based finite-volume multi-resolution Hermite weighted essentially non-oscillatory (MR-HWENO) scheme proposed in our previous article (Li, Shu and Qiu, J. Comput. Phys., 446:110653, 2021) to simulate the steady-state problem. When dealing with the steady-state problem, the process of updating and reconstructing the function values is similar to the previous scheme, but the treatment of the derivative values is changed. To be more specific, instead of evolving in time, in the sense of cell averages, the scheme uses the derivative at the current time step and the function at the next time step to reconstruct the derivative at the next time step by direct linear interpolation. There are two advantages for this approach: the first is its high efficiency, when handling the derivative, neither the update on time nor the calculation of nonlinear weights is required; in the meantime, the CFL number can still be taken up to 0.6 as in the original scheme; the second is its strong convergence, the corresponding average residual can quickly converge to machine accuracy, thus obtaining the desired steady-state solution. One- and two-dimensional numerical experiments are given to verify the high efficiency and strong convergence of the proposed MR-HWENO scheme for the steady-state problems.

This paper presents a family of novel relaxation exponential Runge-Kutta methods for semilinear partial differential equations with dissipative/conservative energy. The novel methods are developed by using the relaxation idea and adding a well-designed governing equation to explicit exponential Runge-Kutta methods. It is shown that the proposed methods can be of high-order accuracy and energy-stable/conserving with mild time step restrictions. In contrast, the previous explicit exponential-type methods are not energy-conserving. Several numerical experiments on KdV equations, Schrödinger equations and Navier-Stokes equations are carried out to illustrate the effectiveness and high efficiency of the methods.

Designing efficient and high-accuracy numerical methods for complex dynamic incompressible Magnetohydrodynamics (MHD) equations remains a challenging problem in various analysis and design tasks. This is mainly due to the nonlinear coupling of the magnetic and velocity fields occurring with convection and Lorentz forces, and multiple physical constraints, which will lead to the limitations of numerical computation. In this paper, we develop the MHDnet as a physics-preserving learning approach to solve MHD problems, where three different mathematical formulations are considered and named $B$ formulation, $A_1$ formulation, and $A_2$ formulation. Then the formulations are embedded into the MHDnet that can preserve the underlying physical properties and divergence-free condition. Moreover, MHDnet is designed by the multi-modes feature merging with multiscale neural network architecture, which can accelerate the convergence of the neural networks (NN) by alleviating the interaction of magnetic fluid coupling across different frequency modes. Furthermore, the pressure fields of three formulations, as the hidden state, can be obtained without extra data and computational cost. Several numerical experiments are presented to demonstrate the performance of the proposed MHDnet compared with different NN architectures and numerical formulations. In future work, we will develop possible applications of inverse problems for coupled equation systems based on the framework proposed in this paper.

Numerical simulation of wetting or dewetting processes is challenging due to the multiscale feature of the moving contact line. This paper presents a numerical method to simulate three-dimensional wetting processes in the framework of lubrication equation and Navier slip condition. A mesoscopic model of the moving contact line is implemented with a cutoff of the computational domain at a small distance from the contact line, where boundary conditions derived from the asymptotic solution of the intermediate region are imposed. This procedure avoids the high resolution required by the local interface near the contact line and enables the adoption of physically small slip lengths. We employ a finite element method to solve the lubrication equation, combined with an arbitrary Lagrangian-Eulerian method to handle the moving boundaries. The method is validated by examining the spreading or sliding of a liquid drop on the wall. The numerical results agree with available exact solutions and approximate theories.

This paper aims to develop and analyze a comprehensive discretized splitting-up numerical scheme for the Zakai equation. This equation arises from a nonlinear filtering problem, where observations incorporate noise modeled by point processes and Wiener processes. Initially, we introduce a regularization parameter and employ a splitting-up approach to break down the Zakai equation into two stochastic differential equations and a partial differential equation (PDE). Subsequently, we employ a finite difference scheme for the temporal dimension and the spectral Galerkin method for the spatial dimension to achieve full discretization of these equations. This results in a numerical solution for the Zakai equation using the splitting-up technique. We demonstrate that this numerical solution converges to the exact solution with a convergence order of $\frac{1}{2}.$ Additionally, we conduct several numerical experiments to illustrate and validate our theoretical findings.

This paper is devoted to a new neural particle method (NPM) based on physics-informed neural networks (PINNs) for modeling free surface flows. Utilizing interface tracking techniques and machine learning (ML) modeling, the new NPM approach with interface tracking and adaptive particle refinement (NPM-LA) is suggested. This method encompasses properties of tracking the interface particles and ensuring the preservation of the designated distribution pattern for interior fluid (computational) particles. The determination of the corresponding physical quantities at these particles is accomplished through the process of inference, a distinctive feature facilitated by ML. The proposed NPM-LA effectively provides solutions for both appropriately tracking the morphology of complex flow surfaces and enhancing the accuracy by dynamically redistributing particles into desired patterns within the computational domain. Two testing cases (the 2D Poiseuille flow problem and a rotating square patch of inviscid fluid) are adopted to examine the performance of the proposed NPM-LA method. The applications to experiments of dam break and wave breaking problems are explored for demonstrating the capability of capturing the complex deforming flow surface.

In this paper, we derive a dimensionless model for compressible multi-component two-phase flows with Peng-Robinson equation of state (EoS), incorporated with the multi-component Navier boundary condition (MNBC). We propose three linearly decoupled and energy-stable numerical schemes for this model. These schemes are developed based on the Lagrange multiplier approach for bulk Helmholtz free energy and surface free energy. One of them is based on a component-wise approach, which requires solving a sequence of linear, separate mass balance equations and leads to an original discrete energy that unconditionally dissipates. Another numerical scheme is based on a component-separate approach, which requires solving a sequence of linear, separate mass balance equations but leads to a modified discrete energy dissipating under certain conditions. Numerical results are presented to verify the effectiveness of the proposed methods.

Using neural networks to solve partial differential equations (PDEs) is gaining popularity as an alternative approach in the scientific computing community. Neural networks can integrate different types of information into the loss function. These include observation data, governing equations, and variational forms, etc. These loss functions can be broadly categorized into two types: observation data loss directly constrains and measures the model output, while other loss functions indirectly model the performance of the network, which can be classified as model loss. However, this alternative approach lacks a thorough understanding of its underlying mechanisms, including theoretical foundations and rigorous characterization of various phenomena. This work focuses on investigating how different loss functions impact the training of neural networks for solving PDEs. We discover a stable loss-jump phenomenon: when switching the loss function from the data loss to the model loss, which includes different orders of derivative information, the neural network solution significantly deviates from the exact solution immediately. Further experiments reveal that this phenomenon arises from the different frequency preferences of neural networks under different loss functions. We theoretically analyze the frequency preference of neural networks under model loss. This loss-jump phenomenon provides a valuable perspective for examining the underlying mechanisms of neural networks in solving PDEs.

The inability to maintain stress continuity across a contact discontinuity is a well-known limitation of some Godunov-type methods developed for gas when directly employed for hypo-elastic solid simulations. Interestingly, this drawback persists in multi-dimensional computations, even when a genuinely multi-dimensional approximate Riemann solver is utilized. To address this challenge, a genuinely two-dimensional Riemann solver is constructed with the enforcement of stress continuity. Subsequently, a path has been constructed by using the present one-dimensional approximate Riemann solver which ensures the stress continuity. Based upon the established path, a discretization method for stress equation is developed by utilizing the path-conservative DLM (Dal Maso, LeFloch, and Murat) approach. Numerical tests demonstrate that the proposed approximate Riemann solver effectively preserves stress continuity, thereby eliminating nonphysical numerical oscillations.