We consider coupled models for particulate flows, where the disperse phase is made of particles with distinct sizes. We are thus led to a system coupling the incompressible Navier-Stokes equations to the multi-component Vlasov-Fokker-Planck equations. We design an asymptotic-preserving numerical scheme to approximate the system. The scheme is based on suitable implicit treatment of the stiff drag force term as well as the Fokker-Planck operator, and can be formally shown to capture the hydrodynamic limit with time step and mesh size independent of the Stokes number. Numerical examples illustrate the accuracy and asymptotic behavior of the scheme, with several interesting applications.

We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a local Virtual Element basis defined within each polygonal control volume. The basis functions are evaluated as an $L_2$ projection of the virtual basis which remains unknown, along the lines of the Virtual Element Method (VEM). Contrary to the VEM approach, the new basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as employed in DG discretizations. The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes that make use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the novel local virtual basis in space and a one-dimensional Lagrange nodal basis in time. The resulting space-time stiffness matrix is stabilized by an extension of the dof–dof stabilization technique adopted in the VEM framework, hence allowing an element-local space-time Galerkin finite element predictor to be evaluated. The new VEM-DG algorithms are rigorously validated against a series of benchmarks in the context of compressible Euler and Navier–Stokes equations.

The small matrix path integral (SMatPI) method is an efficient numerical approach to simulating the evolution of a quantum system coupled to a harmonic bath. The method relies on a sequence of kernel matrices that defines the non-Markovian dynamics of the quantum system. In the original SMatPI method, these kernels are computed indirectly through the QuAPI method. Instead, we focus on the definition of the kernel matrices and reveal a recurrence relation in these matrices. Using such a relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to be much faster than straightforward computation of the kernel matrices based on their definitions. This algorithm bypasses the step to compute the SMatPI matrices by other path integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numerical experiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPI and SMatPI. In spite of this, our method may indicate some new properties of open quantum systems, and has the potential to be generalized to higher-order numerical schemes.

The 3-D static piezoelectric equations with four unknowns are fundamental in studying physical behavior of piezoelectric materials. It is very challenging to numerically solve the equations efficiently due to the so-called “curse of dimensionality”. In this paper, we propose a sparse grid wavelet Galerkin method to overcome the difficulty, by combining the wavelet Galerkin method and the sparse grid technique. The error analysis is also developed in detail. Both the theoretical analysis and numerical results show the previous method outperforms the usual finite element method.

Advancing electron beam applications require pushing toward the quantum degeneracy limit. Nanoscale structured cathodes are a promising electron source for this regime, but the numerical tools for studying these designs remain limited. A previous paper detailed the implemented of a flat-panel fast-multipole-accelerated boundary element method, which solves the relevant Poisson problem. However, flat panels are inadequate and inefficient for representing curved surfaces at the high precision necessary for many applications. Additionally, the boundary element method has an established numerical instability when evaluated near the domain boundary. To resolve this, a general high-order curvilinear element interpolation and modified quadrature method is developed utilizing a differential algebraic mapping for greater accuracy in the boundary surface representation. The boundary instability effect is mitigated by devising local corrections to the quadrature scheme in the form of Cartesian Taylor expansions. This approach is suitably general, requiring only small modifications for application to other kernels, and can easily be incorporated into a fast multipole accelerated framework. The refined algorithm is evaluated with respect to both accuracy and efficiency using several analytic structures and the performance capacity is highlighted by the capability of accurately determining the field enhancement factor for a single nanotip electron cathode.

Based on Ying’s kernel-free boundary integral (KFBI) method [1], a second-order method for general elliptic partial differential equations (PDEs), this paper develops a GPU-accelerated KFBI method for the heat, wave and Schrödinger equations on the irregular domain. Since the limitation of time steps imposed by CFL conditions in the explicit scheme and the inadequate accuracy generated by the fully implicit scheme for the Laplacian operator, the paper selects a series of second-order time discrete schemes, and the Laplacian operator is split into explicit and implicit mixed ones. The Crank-Nicolson method is used to discretize the heat equation in temporal dimension while the implicit $θ$-scheme is for the wave equation. The Strang splitting method is applied to the Schrödinger equation. After discretizing the temporal dimension implicitly, the heat, wave and Schrödinger equations are transformed into a sequence of elliptic equations. The Laplacian operator on the right-hand side of the elliptic equation is obtained from the numerical scheme instead of being discretized and corrected by the five-point difference method. A Cartesian grid-based KFBI method is used to solve the resulting elliptic equations. The KFBI method is accelerated by the graphics processing unit (GPU) with a parallel Cartesian grid solver, achieving a high degree of parallelism. Numerical results show that the proposed method has a second-order accuracy for the heat, wave, and Schrödinger equations. Additionally, the GPU-accelerated solvers for the three types of time-dependent equations are 30 times faster than CPU-based solvers.

In this work, an inverse averaging finite element method (IAFEM) is developed for solving the size-modified Poisson-Nernst-Planck (SMPNP) equations. Comparing with the classical Poisson-Nernst-Planck (PNP) equations, the SMPNP equations add a nonlinear term to each of the Nernst-Planck (NP) fluxes to describe the steric repulsion which can treat multiple nonuniform particle sizes in simulations. Since the new terms include sums and gradients of ion concentrations, the nonlinear coupling of SMPNP equations is much stronger than that of PNP equations. By introducing a generalized Slotboom transform, each of the size-modified NP equation is transformed into a self-adjoint equation with exponentially behaved coefficient, which has similar simple form to the standard NP equation with the Slotboom transformation. This treatment enables employing our recently developed inverse averaging technique to deal with the exponential coefficients of the reformulated formulations, featured with advantages of numerical stability and flux conservation especially in strong nonlinear and convection-dominated cases. Comparing with previous stabilization methods, the IAFEM developed in this paper can still possess the numerical stability when dealing with convection-dominated problems. And it is more concise and easier to be numerically implemented. Numerical experiments about a model problem with analytic solutions are presented to verify the accuracy and order of IAFEM for SMPNP equations. Studies about the size-effects of a sphere model and an ion channel system are presented to show that our IAFEM is more effective and robust than the traditional finite element method (FEM) when solving SMPNP equations in simulations of biological systems.

In this paper, a parallel domain decomposition method is proposed for solving the fully-mixed Stokes-dual-permeability fluid flow model with Beavers-Joseph (BJ) interface conditions. Three Robin-type boundary conditions and a modified weak formulation are constructed to completely decouple the original problem, not only for the free flow and dual-permeability regions but also for the matrix and microfracture flow fields in the dual-porosity media. We derive the equivalence between the original problem and the decoupled systems with some suitable compatibility conditions, and also demonstrate the equivalence of two weak formulations in different Sobolev spaces. Based on the completely decoupled modified weak formulation, the mesh-independent geometric convergence rate of the proposed iterative parallel algorithm is proved rigorously with some suitable parameters. To carry out the convergence analysis of our proposed algorithm, we utilize an important but general convergence lemma for the steady-state problems. Finding the optimized Robin parameters that can accelerate the convergence of the proposed algorithm is an open problem mentioned inhere. Finally, several numerical experiments are presented to illustrate and validate the exclusive features of our proposed algorithm.