We introduce efficient approaches to construct high order finite difference discretizations for solving partial differential equations, based on a composite grid hierarchy. We introduce a modification of the traditional point clustering algorithm, obtained by adding restrictive parameters that control the minimal patch length and the size of the buffer zone. As a result, a reduction in the number of interfacial cells is observed. Based on a reasonable geometric grid setting, we discuss a general approach for the construction of stencils in a composite grid environment. The straightforward approach leads to an ill-posed problem. In our approach we regularize this problem, and transform it into solving a symmetric system of linear of equations. Finally, a stencil repository has been designed to further reduce computational overhead. The effectiveness of the discretizations is illustrated by numerical experiments on second order elliptic differential equations.

Electromagnetic wave propagation in complex dispersive media is governed by the time dependent Maxwell's equations coupled to equations that describe the evolution of the induced macroscopic polarization. We consider "polydispersive" materials represented by distributions of dielectric parameters in a polarization model. The work focuses on a novel computational framework for such problems involving Polynomial Chaos Expansions as a method to improve the modeling accuracy of the Debye model and allow for easy simulation using the Finite Difference Time Domain (FDTD) method. Stability and dispersion analyses are performed for the approach utilizing the second order Yee scheme in two spatial dimensions.

The high order inverse Lax-Wendroff (ILW) procedure is extended to boundary treatment involving complex geometries on a Cartesian mesh. Our method ensures that the numerical resolution at the vicinity of the boundary and the inner domain keeps the fifth order accuracy for the system of the reactive Euler equations with the two-step reaction model. Shock wave propagation in a tube with an array of rectangular grooves is first numerically simulated by combining a fifth order weighted essentially non-oscillatory (WENO) scheme and the ILW boundary treatment. Compared with the experimental results, the ILW treatment accurately captures the evolution of shock wave during the interactions of the shock waves with the complex obstacles. Excellent agreement between our numerical results and the experimental ones further demonstrates the reliability and accuracy of the ILW treatment. Compared with the immersed boundary method (IBM), it is clear that the influence on pressure peaks in the reflected zone is obviously bigger than that in the diffracted zone. Furthermore, we also simulate the propagation process of detonation wave in a tube with three different widths of wall-mounted rectangular obstacles located on the lower wall. It is shown that the shock pressure along a horizontal line near the rectangular obstacles gradually decreases, and the detonation cellular size becomes large and irregular with the decrease of the obstacle width.

Dislocations are line defects in crystalline materials. The Peierls-Nabarro models are hybrid models that incorporate atomic structure of dislocation core into continuum framework. In this paper, we present a numerical method for a generalized Peierls-Nabarro model for curved dislocations, based on the fast multipole method and the iterative grid redistribution. The fast multipole method enables the calculation of the long-range elastic interaction within operations that scale linearly with the total number of grid points. The iterative grid redistribution places more mesh nodes in the regions around the dislocations than in the rest of the domain, thus increases the accuracy and efficiency. This numerical scheme improves the available numerical methods in the literature in which the long-range elastic interactions are calculated directly from summations in the physical domains; and is more flexible to handle problems with general boundary conditions compared with the previous FFT based method which applies only under periodic boundary conditions. Numerical examples using this method on the core structures of dislocations in Al and Cu and in epitaxial thin films are presented.

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations ($B_{co}$ and $\widetilde{B}_{co}$), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.

Many physical processes are described by elliptic or parabolic partial differential equations. For linear stability problems associated with such equations, the inverse Laplacian provides a very effective preconditioner. In addition, it is also readily available in most scientific calculations in the form of a Poisson solver or an implicit diffusive time step. We incorporate Laplacian preconditioning into the inverse Arnoldi method, using BiCGSTAB to solve the large linear systems. Two successful implementations are described: spherical Couette flow described by the Navier-Stokes equations and Bose-Einstein condensation described by the nonlinear Schrödinger equation.

In this work, we develop a minimum action method (MAM) with optimal linear time scaling, called tMAM for short. The main idea is to relax the integration time as a functional of the transition path through optimal linear time scaling such that a direct optimization of the integration time is not required. The Feidlin-Wentzell action functional is discretized by finite elements, based on which h-type adaptivity is introduced to tMAM. The adaptive tMAM does not require reparametrization for the transition path. It can be applied to dealing with quasi-potential: 1) When the minimal action path is subject to an infinite integration time due to critical points, tMAM with a uniform mesh converges algebraically at a lower rate than the optimal one. However, the adaptive tMAM can recover the optimal convergence rate. 2) When the minimal action path is subject to a finite integration time, tMAM with a uniform mesh converges at the optimal rate since the problem is not singular, and the optimal integration time can be obtained directly from the minimal action path. Numerical experiments have been implemented for both SODE and SPDE examples.

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to the $L^2$ norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in the $L^2$ norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

The method of mapping function was first proposed by Henrick et al. [J. Comput. Phys. 207:542-547 (2005)] to adjust nonlinear weights in [0,1] for the fifth-order WENO scheme, and through which the requirement of convergence order is satisfied and the performance of the scheme is improved. Different from Henrick's method, a concept of piecewise polynomial function is proposed in this study and corresponding WENO schemes are obtained. The advantage of the new method is that the function can have a gentle profile at the location of the linear weight (or the mapped nonlinear weight can be close to its linear counterpart), and therefore is favorable for the resolution enhancement. Besides, the function also has the flexibility of quick convergence to identity mapping near two endpoints of [0,1], which is favorable for improved numerical stability. The fourth-, fifth- and sixth-order polynomial functions are constructed correspondingly with different emphasis on aforementioned flatness and convergence. Among them, the fifth-order version has the flattest profile. To check the performance of the methods, the 1-D Shu-Osher problem, the 2-D Riemann problem and the double Mach reflection are tested with the comparison of WENO-M, WENO-Z and WENO-NS. The proposed new methods show the best resolution for describing shear-layer instability of the Riemann problem, and they also indicate high resolution in computations of double Mach reflection, where only these proposed schemes successfully resolved the vortex-pairing phenomenon. Other investigations have shown that the single polynomial mapping function has no advantage over the proposed piecewise one, and it is of no evident benefit to use the proposed method for the symmetric fifth-order WENO. Overall, the fifth-order piecewise polynomial and corresponding WENO scheme are suggested for resolution improvement.

In this paper, we propose a heat jet approach for atomic simulations at finite temperature. Thermal fluctuations are injected into an atomic subsystem from its boundaries, without modifying the governing equations for the interior domain. More precisely, we design a two way local boundary condition, and take the incoming part of a phonon representation for thermal fluctuation input. In this way, non-thermal wave propagation simulations are effectively performed at finite temperature. We further apply this approach to nonlinear chains with the Morse potential. Chains with model parameters fitted to carbon and gold are simulated at room temperature with fidelity.

Periodic structures involving crossed arrays of cylinders appear as special three-dimensional photonic crystals and cross-stacked gratings. Such a structure consists of a number of layers where each layer is periodic in one spatial direction and invariant in another direction. They are relatively simple to fabricate and have found valuable applications. For analyzing scattering properties of such structures, general computational electromagnetics methods can certainly be used, but special methods that take advantage of the geometric features are often much more efficient. In this paper, an efficient method based on operators mapping electromagnetic field components between two spatial directions is developed to analyze structures with crossed arrays of circular cylinders. The method is much simpler than an earlier method based on similar ideas, and it does not require evaluating slowly converging lattice sums.

In this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.