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.

In this work we discuss the numerical discretization of the time-dependent Maxwell's equations using a fully explicit leap-frog type discontinuous Galerkin method. We present a sufficient condition for the stability and error estimates, for cases of typical boundary conditions, either perfect electric, perfect magnetic or first order Silver-Müller. The bounds of the stability region point out the influence of not only the mesh size but also the dependence on the choice of the numerical flux and the degree of the polynomials used in the construction of the finite element space, making possible to balance accuracy and computational efficiency. In the model we consider heterogeneous anisotropic permittivity tensors which arise naturally in many applications of interest. Numerical results supporting the analysis are provided.

A parallel, high-order direct Discontinuous Galerkin (DDG) method has been developed for solving the three dimensional compressible Navier-Stokes equations on 3D hybrid grids. The most distinguishing and attractive feature of DDG method lies in its simplicity in formulation and efficiency in computational cost. The formulation of the DDG discretization for 3D Navier-Stokes equations is detailed studied and the definition of characteristic length is also carefully examined and evaluated based on 3D hybrid grids. Accuracy studies are performed to numerically verify the order of accuracy using flow problems with analytical solutions. The capability in handling curved boundary geometry is also demonstrated. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results obtained indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable results as the widely used BR2 scheme, clearly demonstrating that the DDG method provides an attractive alternative for solving the 3D compressible Navier-Stokes equations.

An idea of designing oscillation-less and high-resolution hybrid schemes is proposed and several types of hybrid schemes based on this idea are presented on block-structured grids. The general framework, for designing various types of hybrid schemes, is established using a Multi-dimensional Optimal Order Detection (MOOD) method proposed by Clain, Diot and Loubère [1]. The methodology utilizes low dissipation or dispersion but less robust schemes to update the solution and then implements robust and high resolution schemes to deal with problematic situations. A wide range of computational methods including central scheme, MUSCL scheme, linear upwind scheme and Weighted Essentially Non Oscillatory (WENO) scheme have been applied in the current hybrid schemes framework. Detailed numerical studies on classical test cases for the Euler system are performed, addressing the issues of the resolution and non-oscillatory property around the discontinuities.

This paper is concerned with numerical methods for the Navier-Stokes-Nernst-Planck-Poisson equation system. The main goal is to construct and analyze some stable time stepping schemes for the time discretization and use a spectral method for the spatial discretization. The main contribution of the paper includes: 1) an useful stability inequality for the weak solution is derived; 2) a first order time stepping scheme is constructed, and the non-negativity of the concentration components of the discrete solution is proved. This is an important property since the exact solution shares the same property. Moreover, the stability of the scheme is established, together with a stability condition on the time step size; 3) a modified first order scheme is proposed in order to decouple the calculation of the velocity and pressure in the fluid field. This new scheme equally preserves the non-negativity of the discrete concentration solution, and is stable under a similar stability condition; 4) a stabilization technique is introduced to make the above mentioned schemes stable without restriction condition on the time step size; 5) finally we construct a second order finite difference scheme in time and spectral discretization in space. The numerical tests carried out in the paper show that all the proposed schemes possess some desirable properties, such as conditionally/unconditionally stability, first/second order convergence, non-negativity of the discrete concentrations, and so on.

In this study an explicit Finite Difference Method (FDM) based scheme is developed to solve the Maxwell's equations in time domain for a lossless medium. This manuscript focuses on two unique aspects – the three dimensional time-accurate discretization of the hyperbolic system of Maxwell equations in three-point non-staggered grid stencil and it's application to parallel computing through the use of Graphics Processing Units (GPU). The proposed temporal scheme is symplectic, thus permitting conservation of all Hamiltonians in the Maxwell equation. Moreover, to enable accurate predictions over large time frames, a phase velocity preserving scheme is developed for treatment of the spatial derivative terms. As a result, the chosen time increment and grid spacing can be optimally coupled. An additional theoretical investigation into this pairing is also shown. Finally, the application of the proposed scheme to parallel computing using one Nvidia K20 Tesla GPU card is demonstrated. For the benchmarks performed, the parallel speedup when compared to a single core of an Intel i7-4820K CPU is approximately 190x.

The superconvergent property of the Discontinuous Galerkin (DG) method for linear hyperbolic systems of partial differential equations in one dimension is explained by relating the DG method to a particular continuous method, whose accuracy depends in part on a local analysis, and in part on information transferred from upwind elements.

Plasma waves with frequencies close to the particular gyrofrequencies of the charged particles in the plasma lose energy due to cyclotron damping. We briefly discuss the gyro-resonance of low frequency plasma waves and ions particularly with regard to particle-in-cell (PiC) simulations. A setup is outlined which uses artificially excited waves in the damped regime of the wave mode's dispersion relation to track the damping of the wave's electromagnetic fields. Extracting the damping rate directly fromthe field data in real or Fourier space is an intricate and non-trivial task. We therefore present a simple method of obtaining the damping rate Γ from the simulation data. This method is described in detail, focusing on a step-by-step explanation of the course of actions. In a first application to a test simulation we find that the damping rates obtained from this simulation generally are in good agreement with theoretical predictions. We then compare the results of one-, two- and three-dimensional simulation setups and simulations with different physical parameter sets.

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyperbolic systems in nonconservative form—the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions.

In this paper, a novel second-order two-scale (SOTS) computational method is developed for nonlinear dynamic thermo-mechanical problems of composites with cylindrical periodicity. The non-linearities of these multi-scale problems were caused by the temperature-dependent properties of the composites. Firstly, the formal SOTS solutions for these problems are constructed by the multiscale asymptotic analysis. Then we theoretically explain the importance of the SOTS solutions by the error analysis in the pointwise sense. In addition, a SOTS numerical algorithm is proposed in detail to effectively solve these problems. Finally, some numerical examples verify the feasibility and effectiveness of the SOTS numerical algorithm we proposed.

In this paper we generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed in [38] for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion [11] while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.

The focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as the activator-activator and short-range inhibition; long-range activation.

In this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.

This paper is devoted to the study of an inverse problem containing a semilinear integrodifferential parabolic equation with an unknown memory kernel. This equation is accompanied by a Robin boundary condition. The missing kernel can be recovered from an additional global measurement in integral form. In this contribution, an error analysis is performed for a time-discrete numerical scheme based on Backward Euler's Method. The theoretical results are supported by some numerical experiments.

Recently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler's elastica image denoising model. We detail the procedures of finding the saddle-points of the augmented Lagrangian functional. Instead of solving associated linear systems by FFT or linear iterative methods (e.g., the Gauss-Seidel method), we adopt a linearized strategy to get an iteration sequence so as to reduce computational cost. In addition, we give some simple complexity analysis for the proposed method. Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method, and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.

A Lagrangian surface hopping algorithm is implemented to study the two dimensional massless Dirac equation for Graphene with an electrostatic potential, in the semiclassical regime. In this problem, the crossing of the energy levels of the system at Dirac points requires a particular treatment in the algorithm in order to describe the quantum transition—characterized by the Landau-Zener probability— between different energy levels. We first derive the Landau-Zener probability for the underlying problem, then incorporate it into the surface hopping algorithm. We also show that different asymptotic models for this problem derived in [O. Morandi, F. Schurrer, J. Phys. A:Math. Theor. 44 (2011) 265301]may give different transition probabilities. We conduct numerical experiments to compare the solutions to the Dirac equation, the surface hopping algorithm, and the asymptotic models of [O. Morandi, F. Schurrer, J. Phys. A: Math. Theor. 44 (2011) 265301].

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B=0) of its magnetic field B. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

Mean curvature-based image registration model firstly proposed by Chumchob-Chen-Brito (2011) offered a better regularizer technique for both smooth and nonsmooth deformation fields. However, it is extremely challenging to solve efficiently this model and the existing methods are slow or become efficient only with strong assumptions on the smoothing parameter β. In this paper, we take a different solution approach. Firstly, we discretize the joint energy functional, following an idea of relaxed fixed point is implemented and combine with Gauss-Newton scheme with Armijo's Linear Search for solving the discretized mean curvature model and further to combine with a multilevel method to achieve fast convergence. Numerical experiments not only confirm that our proposed method is efficient and stable, but also it can give more satisfying registration results according to image quality.

In this article, by applying the Stokes projection and an orthogonal projection with respect to curl and div operators, some new error estimates of finite element method (FEM) for the stationary incompressible magnetohydrodynamics (MHD) are obtained. To our knowledge, it is the first time to establish the error bounds which are explicitly dependent on the Reynolds numbers, coupling number and mesh size. On the other hand, The uniform stability and convergence of an Oseen type finite element iterative method for MHD with respect to high hydrodynamic Reynolds number Re and magnetic Reynolds number Rm, or small δ=1–σ with

(C0, C1 are constants depending only on Ω and F is related to the source terms of equations) are analyzed under the condition that . Finally, some numerical tests are presented to demonstrate the effectiveness of this algorithm.

In this paper the space variable-order fractional Schrödinger equation (VOFSE) is studied numerically, where the variable-order fractional derivative is described here in the sense of the quantum Riesz-Feller definition. The proposed numerical method is the weighted average non-standard finite difference method (WANSFDM). Special attention is given to study the stability analysis and the convergence of the proposed method. Finally, two numerical examples are provided to show that this method is reliable and computationally efficient.