We propose a boundary value correction method for the Brezzi-Douglas-Marini mixed finite element discretization of the Darcy flow with non-homogeneous Neumann boundary condition on 2D curved domains. The discretization is defined on a body-fitted triangular mesh, i.e. the boundary nodes of the mesh lie on the curved physical boundary. However, the boundary edges of the triangular mesh, which are straight, may not coincide with the curved physical boundary. A boundary value correction technique is then designed to transform the Neumann boundary condition from the physical boundary to the boundary of the triangular mesh. One advantage of the boundary value correction method is that it avoids using curved mesh elements and thus reduces the complexity of implementation. We prove that the proposed method reaches optimal convergence for arbitrary order discretizations. Supporting numerical results are presented.

The kernel-free boundary integral (KFBI) method has successfully solved partial differential equations (PDEs) on irregular domains. Diverging from traditional boundary integral methods, the computation of boundary integrals in KFBI is executed through the resolution of equivalent simple interface problems on Cartesian grids, utilizing fast algorithms. While existing implementations of KFBI methods predominantly utilize CPU platforms, GPU architecture’s superior computational capabilities and extensive memory bandwidth offer an efficient resolution to computational bottlenecks. This paper delineates the algorithms adapted for both single-GPU and multiple-GPU applications. On a single GPU, assigning individual threads can control correction, interpolation, and jump calculations. The algorithm is expanded to multiple GPUs to enhance the processing of larger-scale problems. The arrowhead decomposition method is employed in multiple-GPU settings, ensuring optimal computational efficiency and load balancing. Numerical examples show that the proposed algorithm is second-order accurate and efficient. Single-GPU solver runs 50-200 times faster than traditional CPU, and the parallel efficiency of a multiple-GPU distributed solver within the same NUMA node reaches up to 80%.

With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with Lévy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both $α$-stable Lévy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with Lévy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.

Spurious velocities arising from the imperfect offset of the undesired term at the discrete level are frequently observed in numerical simulations of equilibrium multiphase flow systems using the lattice Boltzmann equation (LBE) method. To capture the physical equilibrium state of two-phase fluid systems and eliminate spurious velocities, a well-balanced LBE model based on the incompressible phase-field theory is developed. In this model, the equilibrium distribution function for the Cahn-Hilliard (CH) equation is designed by treating the convection term as a source to avoid the introduction of undesired terms, enabling achievement of possible discrete force balance. Furthermore, this approach allows for the attainment of a divergence-free velocity field, effectively mitigating the impact of artificial compression effects and enhancing numerical stability. Numerical tests, including a flat interface problem, a stationary droplet, and the coalescence of two droplets, demonstrate the well-balanced properties and improvements in the stability of the present model.

In this paper, we propose high order numerical methods to solve a 2D advection diffusion equation, in the highly oscillatory regime. We use an integrator strategy that allows the construction of arbitrary high-order schemes leading to an accurate approximation of the solution without any time step-size restriction. This paper focuses on the multiscale challenges in time of the problem, that come from the velocity, an $ε$-periodic function, whose expression is explicitly known. $ε$–uniform third order in time numerical approximations are obtained. For the space discretization, this strategy is combined with high order finite difference schemes. Numerical experiments show that the proposed methods achieve the expected order of accuracy, and it is validated by several tests across diverse domains and boundary conditions. The novelty of the paper consists of introducing a numerical scheme that is high order accurate in space and time, with a particular attention to the dependency on a small parameter in the time scale. The high order in space is obtained enlarging the interpolation stencil already established in [44], and further refined in [46], with a special emphasis on the squared boundary, especially when a Dirichlet condition is assigned. In such case, we compute an ad hoc Taylor expansion of the solution to ensure that there is no degradation of the accuracy order at the boundary. On the other hand, the high accuracy in time is obtained extending the work proposed in [19]. The combination of high-order accuracy in both space and time is particularly significant due to the presence of two small parameters—$δ$ and $ε$—in space and time, respectively.

This study introduces a methodology for forecasting accelerator performance in Particle Physics algorithms. Accelerating applications can require significant engineering effort, prototyping and measuring the speedup that might finally result in disappointing accelerator performance. The proposed methodology involves performance modelling and forecasting, enabling the prediction of potential speedup, identification of promising acceleration candidates, prior to any significant programming investment. By predicting worst-case scenarios, the methodology assists developers in deciding whether an application can benefit from acceleration, thus optimising effort. A Monte Carlo simulation example demonstrates the effectiveness of the proposed methodology. The result shows that the methodology provides a reasonable estimate for GPUs and, in the context of FPGAs, the predictions are extremely accurate, within 2% of the realised execution time.

Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers, which typically require significant effort, and the presence of nonlinearity in the equation makes such designs even more challenging. Emerging deep learning-based methods for solving PDEs, with the physics-informed neural networks (PINNs) method as a representative, still face significant challenges when directly used to solve PDEs on unbounded domains. Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs. In light of this, this paper proposes a novel and effective data generation-based operator learning method for solving PDEs on unbounded domains. The key idea behind this method is to generate high-quality training data. Specifically, we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term. Then, using these constructed data comprising exact solutions, initial conditions, and source terms, we train an operator learning model called MIONet, which is capable of handling multiple inputs, to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest. Finally, we utilize the generalization ability of this model to predict the solution of the target PDE. The effectiveness of this method is exemplified by solving the wave equation and the Schrödinger equation defined on unbounded domains. More importantly, the proposed method can deal with nonlinear problems, which has been demonstrated by solving Burgers’ equation and Korteweg-de Vries (KdV) equation. The code is available at https://github.com/ZJLAB-AMMI/DGOL.

This paper analyzes a time variable filter approach for the nonlinear fluid-fluid interaction problem. The method applies a time variable filter to improve numerical solution of a variable time step Euler scheme with explicit second-order extrapolation treatment for nonlinear convection and interface terms. Compared with classical second-order method in time, the proposed approach improves time accuracy from the first order to the second order by adding several lines to the code of variable time step Euler scheme. Theoretically, we prove the unconditional energy stability, local $H^1$ stability and error estimates. Numerically, some numerical experiments are provided to test the theoretical results, which illustrate the accuracy and efficiency of the presented method.

Consider the inverse problem of time-harmonic acoustic scattering by an unbounded locally rough interface with bounded obstacles embedded in the lower half-space. An extended reverse time migration (RTM) is proposed to simultaneously reconstruct the locally rough interface and embedded obstacles. By constructing a modified Helmholtz-Kirchhoff identity associated with a planar interface and a mixed reciprocity relation, we propose two new imaging functionals with using both the near-field and far-field measurements. It is shown that the imaging functionals always peak on the local perturbation of the interface and the embedded obstacle. Thus, the two imaging functional can be used to reconstruct the location and shape of the rough surface and the embedded obstacle. Numerical examples are presented to demonstrate the effectiveness of the method.