We develop a general distributed implementation of an adaptive fast multipole method in three space dimensions. We rely on a balanced type of adaptive space discretization which supports a highly transparent and fully distributed implementation. A complexity analysis indicates favorable scaling properties and numerical experiments on up to 512 cores and 1 billion source points verify them. The parameters controlling the algorithm are subject to in-depth experiments and the performance response to the input parameters implies that the overall implementation is well-suited to automated tuning.

Time-domain acoustic scattering problems in two dimensions are studied. The numerical scheme relies on the use of the Convolution Quadrature (CQ) method to reduce the time-domain problem to the solution of frequency-domain Helmholtz equations with complex wavenumbers. These equations are solved with the method of fundamental solutions (MFS), which approximates the solution by a linear combination of fundamental solutions defined at source points inside (outside) the scatterer for exterior (interior) problems. Numerical results show that the coupling of both methods works efficiently and accurately for multistep and multistage based CQ.

The radiative transfer equation is a fundamental equation in transport theory and applications, which is a 5-dimensional PDE in the stationary one-velocity case, leading to great difficulties in numerical simulation. To tackle this bottleneck, we first use the discrete ordinate technique to discretize the scattering term, an integral with respect to the angular variables, resulting in a semi-discrete hyperbolic system. Then, we make the spatial discretization by means of the discontinuous Galerkin (DG) method combined with the sparse grid method. The final linear system is solved by the block Gauss-Seidal iteration method. The computational complexity and error analysis are developed in detail, which show the new method is more efficient than the original discrete ordinate DG method. A series of numerical results are performed to validate the convergence behavior and effectiveness of the proposed method.

In this paper, we reveal a mechanism behind a false accuracy verification encountered with unstructured-grid schemes based on solution reconstruction such as UMUSCL. Third- (or higher-) order of accuracy has been reported for the Euler equations in the literature, but UMUSCL is actually second-order accurate at best for nonlinear equations. False high-order convergence occurs generally for a scheme that is high order for linear equations but second-order for nonlinear equations. It is caused by unexpected linearization of a target nonlinear equation due to too small of a perturbation added to an exact solution used for accuracy verification. To clarify the mechanism, we begin with a proof that the UMUSCL scheme is third-order accurate only for linear equations. Then, we derive a condition under which the third-order truncation error dominates the second-order error and demonstrate it numerically for Burgers’ equation. Similar results are shown for the Euler equations, which disprove some accuracy verification results in the literature. To be genuinely third-order, UMUSCL must be implemented with flux reconstruction.

The paper presents a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other low-degree and nonconforming finite element schemes, the scheme inherits the continuous bilinear form which does not need extra stabilizations and is thus simple to implement.

In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed schemes are constructed based on an auxiliary variable reformulation of the underlying equations, recently introduced by Li et al. [20]. Our objective is to construct and analyze improved schemes, which overcome some of the shortcomings of the existing schemes. In particular, our new schemes have the capability to capture steady solutions for large Reynolds numbers and time step sizes, while keeping the error analysis available. The novelty of our method is twofold: i) Use the Uzawa algorithm to decouple the pressure and the velocity. This is to replace the pressure-correction method considered in [20]. ii) Inspired by the paper [21], we modify the algorithm using an ingredient to capture stationary solutions. In all cases we analyze a first- and second-order schemes and prove the unconditionally energy stability. We also provide an error analysis for the first-order scheme. Finally we validate our schemes by performing simulations of the Kovasznay flow and double lid driven cavity flow. These flow simulations at high Reynolds numbers demonstrate the robustness and efficiency of the proposed schemes.

We present a wavelet-based adaptive method for computing 3D multiscale flows in complex, time-dependent geometries, implemented on massively parallel computers. While our focus is on simulations of flapping insects, it can be used for other flow problems. We model the incompressible fluid with an artificial compressibility approach in order to avoid solving elliptical problems. No-slip and in/outflow boundary conditions are imposed using volume penalization. The governing equations are discretized on a locally uniform Cartesian grid with centered finite differences, and integrated in time with a Runge–Kutta scheme, both of 4th order. The domain is partitioned into cubic blocks with different resolution and, for each block, biorthogonal interpolating wavelets are used as refinement indicators and prediction operators. Thresholding the wavelet coefficients allows to generate dynamically evolving grids, and an adaption strategy tracks the solution in both space and scale. Blocks are distributed among MPI processes and the grid topology is encoded using a tree-like data structure. Analyzing the different physical and numerical parameters allows us to balance their errors and thus ensures optimal convergence while minimizing computational effort. Different validation tests score accuracy and performance of our new open source code, WABBIT. Flow simulations of flapping insects demonstrate its applicability to complex, bio-inspired problems.

This paper develops three high-order accurate discontinuous Galerkin (DG) methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac (NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG (RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff type time discretization (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize the NLD equations and then utilize the explicit multistage high-order Runge-Kutta time discretization for the first-order time derivatives, while the LWDG and TSDG methods, on the contrary, first give the one-stage fourth-order Lax-Wendroff type and the two-stage fourth-order time discretizations of the NLD equations, respectively, and then discretize the first- and higher-order spatial derivatives by using the spatial DG approximation. The $L^2$ stability of the 2D semi-discrete DG approximation is proved in the RKDG methods for a general triangulation, and the computational complexities of three 1D DG methods are estimated. Numerical experiments are conducted to validate the accuracy and the conservation properties of the proposed methods. The interactions of the solitary waves, the standing and travelling waves are investigated numerically and the 2D breathing pattern is observed.

In this article we present two types of nonlinear positivity-preserving finite volume (PPFV) schemes for a class of three-dimensional heat conduction equations on general polyhedral meshes. First, we present a new parameter selection strategy on the one-sided flux and establish a nonlinear PPFV scheme based on a two-point flux with higher efficiency. By comparing with the scheme proposed in [H. Xie, X. Xu, C. Zhai, H. Yong, Commun. Comput. Phys. 24 (2018) 1375–1408], our scheme avoids the assumption that the values of auxiliary unknowns are nonnegative, which makes our interpolation formulae suitable to be constructed by existing approaches with high accuracy and well robustness (e.g., the finite element method), thus enhancing the adaptability to distorted meshes with large deformations. Then we derive a linear multi-point flux involving combination coefficients and, via the Patankar trick, obtain another nonlinear PPFV scheme that is concise and easy to implement. The selection strategy of combination coefficients is also provided to improve the convergence behavior of the Picard procedure. Furthermore, the existence and positivity-preserving properties of these two nonlinear PPFV solutions are proved. Numerical experiments with the discontinuous diffusion scalar as well as discontinuous and anisotropic diffusion tensors are given to confirm our theoretical findings and demonstrate that our schemes both can achieve ideal-order accuracy even on severely distorted meshes.

Quantum computation using optical modes has been well-established in its ability to construct deep neural networks. These networks have been shown to be flexible both architecturally as well as in terms of the type of data being processed. We leverage this property of the Continuous-Variable (CV) model to construct stacked single mode networks that are shown to learn structured classical information, while placing no restrictions on the size of the network, and at the same time maintaining its complexity. The hallmark of the CV model is its ability to forge non-linear functions using a set of gates that allows it to remain completely unitary. The proposed model exemplifies that the appropriate photonic hardware can be integrated with present day optical communication systems to meet our information processing requirements. In this paper, using the Strawberry Fields software library on the MNIST dataset of hand-written digits, we demonstrate the adaptability of the network to learn classical information in a multitude of machine learning tasks to very large fidelities.

In this paper we introduce EMPIRE-PIC, a finite element method particle-in-cell (FEM-PIC) application developed at Sandia National Laboratories. The code has been developed in C++ using the Trilinos library and the Kokkos Performance Portability Framework to enable running on multiple modern compute architectures while only requiring maintenance of a single codebase. EMPIRE-PIC is capable of solving both electrostatic and electromagnetic problems in two- and three-dimensions to second-order accuracy in space and time. In this paper we validate the code against three benchmark problems — a simple electron orbit, an electrostatic Langmuir wave, and a transverse electromagnetic wave propagating through a plasma. We demonstrate the performance of EMPIRE-PIC on four different architectures: Intel Haswell CPUs, Intel's Xeon Phi Knights Landing, ARM Thunder-X2 CPUs, and NVIDIA Tesla V100 GPUs attached to IBM POWER9 processors. This analysis demonstrates scalability of the code up to more than two thousand GPUs, and greater than one hundred thousand CPUs.