An iterative solver based on the immersed interface method is proposed to solve the pressure in a two-fluid flow on a Cartesian grid with second-order accuracy in the infinity norm. The iteration is constructed by introducing an unsteady term in the pressure Poisson equation. In each iteration step, a Helmholtz equation is solved on the Cartesian grid using FFT. The combination of the iteration and the immersed interface method enables the solver to handle various jump conditions across two-fluid interfaces. This solver can also be used to solve Poisson equations on irregular domains.

We propose a fully conservative and less oscillatory multi-moment scheme for the approximation of hyperbolic conservation laws. The proposed scheme (CIP-CSL3ENO) is based on two CIP-CSL3 schemes and the essentially non-oscillatory (ENO) scheme. In this paper, we also propose an ENO indicator for the multimoment framework, which intentionally selects non-smooth stencil but can efficiently minimize numerical oscillations. The proposed scheme is validated through various benchmark problems and a comparison with an experiment of two droplets collision/separation. The CIP-CSL3ENO scheme shows approximately fourth-order accuracy for smooth solution, and captures discontinuities and smooth solutions simultaneously without numerical oscillations for solutions which include discontinuities. The numerical results of two droplets collision/separation (3D free surface flow simulation) show that the CIP-CSL3ENO scheme can be applied to various types of fluid problems not only compressible flow problems but also incompressible and 3D free surface flow problems.

In this study, the Lattice Boltzmann Method (LBM) is implemented through a finite-volume approach to perform 2-D, incompressible, and turbulent fluid flow analyses on structured grids. Even though the approach followed in this study necessitates more computational effort compared to the standard LBM (the so called stream and collide scheme), using the finite-volume method, the known limitations of the stream and collide scheme on lattice to be uniform and Courant-Friedrichs-Lewy (CFL) number to be one are removed. Moreover, the curved boundaries in the computational domain are handled more accurately with less effort. These improvements pave the way for the possibility of solving fluid flow problems with the LBM using coarser grids that are refined only where it is necessary and the boundary layers might be resolved better.

We present a three dimensional preconditioned implicit free-surface capture scheme on tetrahedral grids. The current scheme improves our recently reported method [10] in several aspects. Specifically, we modified the original eigensystem by applying a preconditioning matrix so that the new eigensystem is virtually independent of density ratio, which is typically large for practical two-phase problems. Further, we replaced the explicit multi-stage Runge-Kutta method by a fully implicit Euler integration scheme for the Navier-Stokes (NS) solver and the Volume of Fluids (VOF) equation is now solved with a second order Crank-Nicolson implicit scheme to reduce the numerical diffusion effect. The preconditioned restarted Generalized Minimal RESidual method (GMRES) is then employed to solve the resulting linear system. The validation studies show that with these modifications, the method has improved stability and accuracy when dealing with large density ratio two-phase problems.

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function ϕ(x,y). Our adaptive mesh refinement is done within a small tube of |ϕ(x,y)|≤δ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.

The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.

High-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.

Quasi-Monte Carlo methods and stochastic collocation methods based on sparse grids have become popular with solving stochastic partial differential equations. These methods use deterministic points for multi-dimensional integration or interpolation without suffering from the curse of dimensionality. It is not evident which method is best, specially on random models of physical phenomena. We numerically study the error of quasi-Monte Carlo and sparse grid methods in the context of ground-water flow in heterogeneous media. In particular, we consider the dependence of the variance error on the stochastic dimension and the number of samples/collocation points for steady flow problems in which the hydraulic conductivity is a lognormal process. The suitability of each technique is identified in terms of computational cost and error tolerance.

We present a three dimensional model of an open elastic tube immersed in fluid to understand valveless pumping mechanism. A fluid-tube interaction problem is simulated by the volume conserved immersed boundary method which prevents the generation of spurious velocity field near the tube and local cluster of the tube surface. In order to explain pumping phenomena without valves, average net flow is measured by changing parameter values such as pumping frequency, compression duration, and pumping amplitude. Some frequencies that make the system reach maximal or minimal net flow are selected to study case by case. We also study the effectiveness of fluid mixing using the Shannon entropy increase rate.

The equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.

This work pertains to numerical aspects of a finite element method based discontinuous functions. Our study focuses on the Interior Penalty Discontinuous Galerkin method (IPDGM) because of its high-level of flexibility for solving the full wave equation in heterogeneous media. We assess the performance of IPDGM through a comparison study with a spectral element method (SEM). We show that IPDGM is as accurate as SEM. In addition, we illustrate the efficiency of IPDGM when employed in a seismic imaging process by considering two-dimensional problems involving the Reverse Time Migration.

This paper describes an adaptive preconditioner for numerical continuation of incompressible Navier–Stokes flows based on Stokes preconditioning [42] which has been used successfully in studies of pattern formation in convection. The preconditioner takes the form of the Helmholtz operator I–ΔtL which maps the identity (no preconditioner) for Δt≪1 to Laplacian preconditioning for Δt≫1. It is built on a first order Euler time-discretization scheme and is part of the family of matrix-free methods. The preconditioner is tested on two fluid configurations: three-dimensional doubly diffusive convection and a two-dimensional projection of a shear flow. In the former case, it is found that Stokes preconditioning is more efficient for , away from the values used in the literature. In the latter case, the simple use of the preconditioner is not sufficient and it is necessary to split the system of equations into two subsystems which are solved simultaneously using two different preconditioners, one of which is parameter dependent. Due to the nature of these applications and the flexibility of the approach described, this preconditioner is expected to help in a wide range of applications.

In this paper, a conservative parallel iteration scheme is constructed to solve nonlinear diffusion equations on unstructured polygonal meshes. The design is based on two main ingredients: the first is that the parallelized domain decomposition is embedded into the nonlinear iteration; the second is that prediction and correction steps are applied at subdomain interfaces in the parallelized domain decomposition method. A new prediction approach is proposed to obtain an efficient conservative parallel finite volume scheme. The numerical experiments show that our parallel scheme is second-order accurate, unconditionally stable, conservative and has linear parallel speed-up.

In this paper, we present a new type of Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the Hamilton-Jacobi equations on the finite volume framework. The cell averages of the function and its first one (in one dimension) or two (in two dimensions) derivative values are together evolved via time approaching and used in the reconstructions. And the major advantages of the new HWENO schemes are their compactness in the spacial field, purely on the finite volume framework and only one set of small stencils is used for different type of the polynomial reconstructions. Extensive numerical tests are performed to illustrate the capability of the methodologies.

In this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.

The symmetric energy-conserved splitting FDTD scheme developed in is a very new and efficient scheme for computing the Maxwell’s equations. It is based on splitting the whole Maxwell’s equations and matching the x-direction and y-direction electric fields associated to the magnetic field symmetrically. In this paper, we make further study on the scheme for the 2D Maxwell’s equations with the PEC boundary condition. Two new energy-conserved identities of the symmetric EC-S-FDTD scheme in the discrete H1-norm are derived. It is then proved that the scheme is uncondi-tionally stable in the discrete H1-norm. By the new energy-conserved identities, the super-convergence of the symmetric EC-S-FDTD scheme is further proved that it is of second order convergence in both time and space steps in the discrete H1-norm. Numerical experiments are carried out and confirm our theoretical results.

The implicit 2D3V particle-in-cell (PIC) code developed to study the interaction of ultrashort pulse lasers with matter [G. M. Petrov and J. Davis, Computer Phys. Comm. 179, 868 (2008); Phys. Plasmas 18, 073102 (2011)] has been parallelized using MPI (Message Passing Interface). The parallelization strategy is optimized for a small number of computer cores, up to about 64. Details on the algorithm implementation are given with emphasis on code optimization by overlapping computations with communications. Performance evaluation for 1D domain decomposition has been made on a small Linux cluster with 64 computer cores for two typical regimes of PIC operation: “particle dominated”, for which the bulk of the computation time is spent on pushing particles, and “field dominated”, for which computing the fields is prevalent. For a small number of computer cores, less than 32, the MPI implementation offers a significant numerical speed-up. In the “particle dominated” regime it is close to the maximum theoretical one, while in the “field dominated” regime it is about 75-80 % of the maximum speed-up. For a number of cores exceeding 32, performance degradation takes place as a result of the adopted 1D domain decomposition. The code parallelization will allow future implementation of atomic physics and extension to three dimensions.

It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the “pollution error” caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weakform. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.

Hierarchical bases of arbitrary order for (div)-conforming triangular and tetrahedral elements are constructed with the goal of improving the conditioning of the mass and stiffness matrices. For the basis with the triangular element, it is found numerically that the conditioning is acceptable up to the approximation of order four, and is better than a corresponding basis in the dissertation by Sabine Zaglmayr [High Order Finite Element Methods for Electromagnetic Field Computation, Johannes Kepler Universität, Linz, 2006]. The sparsity of the mass matrices from the newly constructed basis and from the one by Zaglmayr is similar for approximations up to order four. The stiffness matrix with the new basis is much sparser than that with the basis by Zaglmayr for approximations up to order four. For the tetrahedral element, it is identified numerically that the conditioning is acceptable only up to the approximation of order three. Compared with the newly constructed basis for the triangular element, the sparsity of the mass matrices from the basis for the tetrahedral element is relatively sparser.

In magnetically confined plasma research, the understandings of small and large perturbations at equilibrium are both critical for plasma controlling and steady state operation. Numerical simulations using original MHD model can hardly give clear picture for small perturbations, while non-conservative perturbed MHD model may break conservation law, and give unphysical results when perturbations grow large after long-time computation. In this paper, we present a nonlinear conservative perturbed MHD model by splitting primary variables in original MHD equations into equilibrium part and perturbed part, and apply an approach in the framework of discontinuous Galerkin (DG) spatial discretization for numerical solutions. This enables high resolution of very small perturbations, and also gives satisfactory non-smooth solutions for large perturbations, which are both broadly concerned in magnetically confined plasma research. Numerical examples demonstrate satisfactory performance of the proposed model clearly. For small perturbations, the results have higher resolution comparing with the original MHD model; for large perturbations, the non-smooth solutions match well with existing references, confirming reliability of the model for instability investigations in magnetically confined plasma numerical research.