arXiv subjects

Wenjun Ying

Publications and source records attributed to Wenjun Ying.

32 records · Page 2Linked to original sources

arXiv2024

A GPU-accelerated Cartesian grid method for PDEs on irregular domain

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 speeds 50-200 times than traditional CPU while the eight GPUs distributed solver yields up to 60% parallel efficiency.

math.NA↗

arXiv2024

PM2D: A parallel GPU-based code for the kinetic simulation of laser plasma instabilities in large scale plasmas

Laser plasma instabilities (LPIs) have significant influences on the laser energy deposition efficiency, hot electron generation, and uniformity of irradiation in inertial confined fusion (ICF). In contrast to theoretical analysis of linear development of LPIs, numerical simulations play a more and more important role in revealing the complex physics of LPIs. Since LPIs are typically a three-wave coupling process, the precise kinetic simulation of LPIs requires to resolve the laser period (around one femtosecond) and laser wavelength (less than one micron). In this paper, a full wave fluid model of LPIs is constructed and numerically solved by the particle-mesh method, where the plasma is described by macro particles that can move across the mesh grids freely. Based upon this model, a two-dimensional (2D) GPU code named PM2D is developed. It can simulate the kinetic effects of LPIs self-consistently as normal particle-in-cell (PIC) codes. Moreover, as the physical model adopted in the PM2D code is specifically constructed for LPIs, the required macro particles per grid in the simulations can be largely reduced and thus overall simulation cost is considerably reduced comparing with typical PIC codes. Moreover, the numerical noise in our PM2D code is much lower, which makes it more robust than PIC codes in the simulation of LPIs for the long-time scale above 10 picoseconds. After the distributed computing is realized, our PM2D code is able to run on GPU clusters with a total mesh grids up to several billions, which meets the typical requirements for the simulations of LPIs at ICF experimental scale with reasonable cost.

physics.plasm-ph↗

arXiv2024

Physics-informed Data-driven Cavitation Model for a Specific MG EOS

We present a novel one-fluid cavitation model of a specific Mie-Grüneisen equation of state(EOS), named polynomial EOS, based on an artificial neural network. Not only the physics-informed equation but also the experimental data are embedded into the proposed model by an optimization problem. The physics-informed data-driven model provides the concerned pressure within the cavitation region, where the density tends to zero when the pressure falls below the saturated pressure. The present model is then applied to computing the challenging compressible multi-phase flow simulation, such as nuclear and underwater explosions. Numerical simulations show that our model in application agrees well with the corresponding experimental data, ranging from one dimension to three dimensions with the $h-$adaptive mesh refinement algorithm and load balance techniques in the structured and unstructured grid.

physics.flu-dyn↗

arXiv2024

A Stabilized Parametric Finite Element Method for Surface Diffusion with an Arbitrary Surface Energy

We proposed a structure-preserving stabilized parametric finite element method (SPFEM) for the evolution of closed curves under anisotropic surface diffusion with an arbitrary surface energy $\hatγ(θ)$. By introducing a non-negative stabilizing function $k(θ)$ depending on $\hatγ(θ)$, we obtained a novel stabilized conservative weak formulation for the anisotropic surface diffusion. A SPFEM is presented for the discretization of this weak formulation. We construct a comprehensive framework to analyze and prove the unconditional energy stability of the SPFEM under a very mild condition on $\hatγ(θ)$. This method can be applied to simulate solid-state dewetting of thin films with arbitrary surface energies, which are characterized by anisotropic surface diffusion and contact line migration. Extensive numerical results are reported to demonstrate the efficiency, accuracy and structure-preserving properties of the proposed SPFEM with anisotropic surface energies $\hatγ(θ)$ arising from different applications.

math.NA↗

arXiv2024

Learning Domain-Independent Green's Function For Elliptic Partial Differential Equations

Green's function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green's function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green's function, referred to as BIN-G. We evaluate the Green's function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green's function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green's function for PDEs with variable coefficients. The learned Green's function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

physics.comp-ph↗

arXiv2023

An Alternating Direction Implicit Method for Mean Curvature Flows

This paper is concerned with the mean curvature flow, which describes the dynamics of a hypersurface whose normal velocity is determined by local mean curvature. We present a Cartesian grid-based method for solving mean curvature flows in two and three space dimensions. The present method embeds a closed hypersurface into a fixed Cartesian grid and decomposes it into multiple overlapping subsets. For each subset, extra tangential velocities are introduced such that marker points on the hypersurface only moves along grid lines. By utilizing an alternating direction implicit (ADI)-type time integration method, the subsets are evolved alternately by solving scalar parabolic partial differential equations on planar domains. The method removes the stiffness using a semi-implicit scheme and has no high-order stability constraint on time step size. Numerical examples in two and three space dimensions are presented to validate the proposed method.

math.NA↗

arXiv2023

Kernel Free Boundary Integral Method for 3D Stokes and Navier Equations on Irregular Domains

A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green's functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

math.NA↗

arXiv2023

Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids

A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete $\ell^2$-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests.

math.NA↗

arXiv2023

Second order convergence of a modified MAC scheme for Stokes interface problems

Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell (MAC) scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi (LBB) condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete $\ell^2$-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.

math.NA↗

arXiv2022

Numerical study on viscous fingering using electric fields in a Hele-Shaw cell

We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate boundary integral method where a coupled integral equation system is formulated. Although the stiffness due to the high order spatial derivatives can be removed, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove this physically imposed stiffness by employing a rescaling scheme, which accelerates the slow dynamics and reduces the computational cost. Our nonlinear results reveal that positive currents restrain finger ramification and promote overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no flux is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law $\displaystyle (t_*-t)^b$, where $t_*$ is the estimated pinch-off time.

physics.flu-dyn↗

arXiv2021

A Kernel-free Boundary Integral Method for the Bidomain Equations

The bidomain equations have been widely used to mathematically model the electrical activity of the cardiac tissue. In this work, we present a potential theory-based Cartesian grid method which is referred as the kernel-free boundary integral (KFBI) method which works well on complex domains to efficiently simulate the linear diffusion part of the bidomain equation. After a proper temporal discretization, the KFBI method is applied to solve the resulting homogeneous Neumann boundary value problems with a second-order accuracy. According to the potential theory, the boundary integral equations reformulated from the boundary value problems can be solved iteratively with the simple Richardson iteration or the Krylov subspace iteration method. During the iteration, the boundary and volume integrals are evaluated by limiting the structured grid-based discrete solutions of the equivalent interface problems at quasi-uniform interface nodes without the need to know the analytical expression of Green's functions. In particular, the discrete linear system of the equivalent interface problem obtained from the standard finite difference schemes or the finite element schemes can be efficiently solved by fast elliptic solvers such as the fast Fourier transform based solvers or those based on geometric multigrid iterations after an appropriate modification at the irregular grid nodes. Numerical results for solving the FitzHugh-Nagumo bidomain equations in both two- and three-dimensional spaces are presented to demonstrate the numerical performance of the KFBI method such as the second-order accuracy and the propagation and scroll wave of the voltage simulated on the real human left ventricle model.

math.NA↗

arXiv2018

Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation can be solved by replacing the integral operator with a finite difference calculation on a regular grid. A practical method of this type has been developed by the second author. In this paper we prove the validity of a simplified version of this method for the Dirichlet problem in a general domain in $R^2$ or $R^3$. Given a boundary value, we solve for a discrete version of the density of the double layer potential using a low order interface method. It produces the Shortley-Weller solution for the unknown harmonic function with accuracy $O(h^2)$. We prove the unique solvability for the density, with bounds in norms based on the energy or Dirichlet norm, using techniques which mimic those of exact potentials. The analysis reveals that this crude method maintains much of the mathematical structure of the classical integral equation. Examples are included.

math.NA↗

arXiv2016

Uniform convergent scheme for strongly anisotropic diffusion equations with closed field lines

In magnetized plasma, the magnetic field confines particles around field lines. The ratio between the intensity of the parallel and perpendicular viscosity or heat conduction may reach the order of $10^{12}$. When the magnetic fields have closed field lines and form a "magnetic island", the convergence order of most known schemes depends on the anisotropy strength. In this paper, by integration of the original differential equation along each closed field line, we introduce a simple but very efficient asymptotic preserving reformulation, which yields uniform convergence with respect to the anisotropy strength. Only slight modification to the original code is required and neither change of coordinates nor mesh adaptation is needed. Numerical examples demonstrating the performance of the new scheme are presented.

math.NA↗

arXiv2016

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

math.NA↗