It is well known since 1960s that by exploring the tensor product structure of the discrete Laplacian on Cartesian meshes, one can develop a simple direct Poisson solver with an $\mathcal{O}(N^{\frac{d+1}{d}})$ complexity in $d$-dimension, where $N$ is the number of the total unknowns. The GPU acceleration of numerically solving PDEs has been explored successfully around fifteen years ago and become more and more popular in the past decade, driven by significant advancement in both hardware and software technologies, especially in the recent few years. We present in this paper a simple but extremely fast MATLAB implementation on a modern GPU, which can be easily reproduced, for solving 3D Poisson type equations using a spectral-element method. In particular, it costs less than one second on a Nvidia A100 for solving a Poisson equation with one billion degree of freedoms. We also present applications of this fast solver to solve a linear (time-independent) Schrödinger equation and a nonlinear (time-dependent) Cahn-Hilliard equation.

In many applications, such as plasma edge simulation of a nuclear fusion reactor, a coupled PDE/kinetic description is required, which is usually solved with a coupled finite-volume/Monte-Carlo method. Different procedures have been proposed to estimate the source terms in the finite volume part that appear from the Monte Carlo part of the simulation. In this paper, we perform a systematic (analytical and numerical) comparison of the variance and computational cost of source term estimations using track-length based simulation techniques and track-length and next-event estimation procedures. We analyze in detail a scenario with forward-backward scattering in a one-dimensional slab. For this test case, we perform a parametric study of the expected statistical error and computational cost, revealing the large differences in performance of the estimation procedures depending on the problem parameters. The analytical part of the comparison is based on an invariant imbedding procedure, in which systems of ordinary differential equations (ODEs) are derived that quantify the statistical error and computational cost of each estimator in a simplified setting.

Simulating physical problems with multi-time scale coupling presents a considerable challenge due to the concurrent solution of processes with different time scales. This complexity arises from the necessity to evolve large time scale processes over long physical time, while simultaneously small time step sizes are required to unveil the underlying physics in shorter time scale processes. To address this inherent conflict in the multi-time scale coupling problems, we propose an explicit multi-time step algorithm within the framework of smoothed particle hydrodynamics (SPH), coupled with a solid dynamic relaxation scheme, to quickly achieve equilibrium state in the comparatively fast solid response process. To assess the accuracy and efficiency of the proposed algorithm, a manuscript torsional example, two distinct scenarios, i.e., a nonlinear hardening bar stretching and a fluid diffusion coupled with Nafion membrane flexure, are simulated. The obtained results exhibit good agreement with analytical solution, outcomes from other numerical methods and experimental data. With this explicitly multi-time step algorithm, the simulation time is reduced firstly by independently addressing different processes being solved under distinct time step sizes, which stands in contrast to the implicit counterpart, and secondly decreasing the simulation time required to achieve a steady state for the solid by incorporating the dynamic relaxation scheme.

In this work, we propose to learn time integration schemes based on neural networks which satisfy three distinct sets of mathematical constraints, i.e., unconstrained, semi-constrained with the root condition, and fully-constrained with both root and consistency conditions. We focus on the learning of 3-step linear multistep methods, which we subsequently applied to solve three model PDEs, i.e., the heat equation, the wave equation, and the Burgers’ equation. The results show that the prediction error of the learned fully-constrained scheme is close to that of the Runge-Kutta method and Adams-Bashforth method. Compared to the traditional methods, the learned unconstrained and semi-constrained schemes significantly reduce the prediction error on coarse grids, with an overall better performance for the semi-constrained model. On a grid $4×$ coarser than the reference, the mean square error (MSE) shows a reduction of up to two order of magnitude for some heat equation cases, and a substantial improvement in phase prediction for the wave equation. On a $32×$ coarser grid, the MSE for the Burgers’ equation can be reduced by up to 40% to 45%. Tests on the two-dimensional heat equation and wave equation show a good generalization capability for the constant optimized coefficients.

This paper introduces a novel data-driven shock-capturing for unstructured finite volume method (FVM) on multi-dimensional compressible flows, aiming at enhancing accuracy in smooth region while ensuring robustness in discontinuous region, by machine-learning models. We achieve this objective via a two-step process based on a tree model and fully-connected neural network (FCNN) models. In the first step, a tree model is employed as a troubled-cell indicator to divide the computational domain into discontinuous and smooth regions. In the second step, we employ FCNN models to reconstruct solution variables at a cell interface, which are used to obtain numerical fluxes. Notably, two FCNN models are used for discontinuous and smooth regions, respectively, with the former designed for robust capturing of discontinuities, and the latter intended to enhance accuracy in smooth region. To train such models, we generate two types of datasets using various analytic functions to mimic discontinuous and smooth FVM solutions. In addition, we define specific input features to provide the information on solution distribution and coordinates, enabling the extension onto unstructured meshes. Finally, we systematically validate the proposed method through a comprehensive set of numerical experiments, ranging from scalar conservation laws, Euler equations to Navier-Stokes equations. Computational experiments demonstrate enhanced performance in terms of robustness and accuracy, highlighting the capability of data-driven reconstruction in complex flow simulations.

Pressure dislocation might be observed when the modified ghost fluid method (MGFM) is applied to simulate radially symmetric multi-medium fluid flows for a long time. We disclose the insightful reason that the MGFM cannot satisfy balanced boundary conditions when applied to simulate the radially symmetric compressible multi-medium flows, resulting in a first-order temporal error in the interface region. To impose these balanced boundary conditions, we develop an improved MGFM in this work. The reconstruction for the initial value of the multi-medium Riemann problem at the interface is specially designed in combination with these balanced boundary conditions. In addition, the predicted instantaneous interface states and balanced boundary conditions are then utilized to define the ghost fluid states. Theoretical analysis shows that the improved MGFM can satisfy these balanced boundary conditions and effectively eliminate the first-order temporal error at the interface. Its extension to two dimensions is also presented. Numerical results show that the proposed improved MGFM can restraint pressure dislocation and overheating at the material interface very well and effectively improve mass conservation.

Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as a large number of neighborhood query operations being invoked on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on the combinatorial map, and an efficient parallel algorithm is also designed for the assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.

The identification and visualization of Lagrangian structures in flows plays a crucial role in the study of dynamic systems and fluid dynamics. The Finite Time Lyapunov Exponent (FTLE) has been widely used for this purpose; however, it only approximates the flow by considering the positions of particles at the initial and final times, ignoring the actual trajectory of the particle. To overcome this limitation, we propose a novel quantity that extends and generalizes the FTLE by incorporating trajectory metrics as a measure of similarity between trajectories. Our proposed method utilizes trajectory metrics to quantify the distance between trajectories, providing a more robust and accurate measure of the LCS. By incorporating trajectory metrics, we can capture the actual path of the particle and account for its behavior over time, resulting in a more comprehensive analysis of the flow. Our approach extends the traditional FTLE approach to include trajectory metrics as a means of capturing the complexity of the flow.