Arbitrary high order finite-volume methods for electromagnetic wave propagation

Problems in electromagnetic wave propagation often require high accuracy approximations with low resolution computational grids. For non-stationary problems such schemes should possess the same approximation order in space and time. In the present article we propose for electromagnetic applications an explicit class of robust finite-volume (FV) schemes for the Maxwell equations. To achieve high accuracy we combine the FV method with the so-called ADER approach resulting in schemes which are arbitrary high order accurate in space and time. Numerical results and convergence investigations are shown for two and three-dimensional test cases on Cartesian grids, where the used FV-ADER schemes are up to 8th order accurate in both space and time.     

Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton–Jacobi–Bellman Equations

We analyse two practical aspects that arise in the numerical solution of Hamilton–Jacobi–Bellman equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature.     

Numerical errors resulting from finite-difference approximation in computations of a one-dimensional Schrödinger equation with the Trotter formula

The parallel algorithm for solving time-dependent Schrödinger equations devised by De Raedt and based on the Trotter formula is not only simple but also unconditionally stable, explicit, and local. We consider the numerical errors resulting from the finite-difference approximation of De Raedt's algorithm by comparing an exact solution of a free particle with the approximate solution calculated by using the Trotter formula, which depends on the size of the spatial-temporal lattice.     

Truncation versus mapping in the spectral approximation to the Korteweg-De Vries equation

We propose two different approaches to the numerical solution of the initial boundary value problem for the Korteweg-De Vries equation; the former is based on the truncation of the domain, the latter on the reduction of the real axis to a bounded interval by a suitable mapping technique. In both cases we consider spectral Chebyshev collocation methods for the space discretization and finite difference schemes for advancing in time. Both single and multi-domain approaches are discussed. We report numerical experiments showing the stability and convergence properties of the methods     

Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k²+k²h²+h²) and O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

Chebyshev rational spectral method for long-short wave equations

We consider the initial boundary value problem of the long-short wave equations on the whole line. A fully discrete spectral approximation scheme is developed based on Chebyshev rational functions in space and central difference in time. A priori estimates are derived which are crucial to study numerical stability and convergence of the fully discrete scheme. Then, unconditional numerical stability is proved. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented to demonstrate the efficiency and accuracy of the convergence results.     

Family of quadratic spline difference schemes for a convection–diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.     

Stability of Difference Splitting Schemes for Convection Diffusion Equation

We consider numerical modeling of the propagation of pollution in the air on the basis of geometrical splitting method for three-dimensional nonstationary convection diffusion equations. Splitting difference schemes in the form of explicit computing schemes are proposed to solve the obtained one-dimensional problems. The approximation, monotonicity, and stability of the proposed difference schemes are investigated.     

Single and multidomain Chebyshev collocation methods for the Korteweg-de Vries equation

We propose spectral Chebyshev collocation algorithms for the approximation of the initial and boundary value problem for the Korteweg-de Vries equation. Both single and multidomain approaches are discussed. Different methods for the treatment of the boundary conditions are considered. The numerical analysis of the eigenvalues' behaviour of the spectral differentiation operators involved in the approximation suggests appropriate finite difference methods for time-marching. Several numerical experiments have been performed, which prove spectral convergence and stability of the proposed schemes.     

On the evaluation of correction terms in Gauss-Legendre quadrature

In the numerical integration of analytic functions, the singularities of the integrand affect the rate of convergence of the quadrature. This convergence can be improved significantly by adding the residue correction terms for the poles of the integrand. But this needs the evaluation of the basis function and its corresponding second kind function with complex arguments. We indicate a simple and accurate method to evaluate the correction term involving the basis and its second kind functions in the case of Gauss-Legendre quadrature. This approach does not call for the evaluation of the hypergeometric functions.     

Pseudospectral method of solution of the Fitzhugh–Nagumo equation

We present a study of the convergence of different numerical schemes in the solution of the Fitzhugh–Nagumo equations in the form of two coupled reaction diffusion equations for activator and inhibitor variables. The diffusion coefficient for the inhibitor is taken to be zero. The Fitzhugh–Nagumo equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are a Chebyshev multidomain method, a finite difference method and the method developed by Barkley [D. Barkley, A model for fast computer simulation of excitable media, Physica D, 49 (1991) 61–70]. We consider two different models for the local dynamics. We present results for plane wave propagation in one dimension and spiral waves for two dimensions. We use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. Zero flux boundary conditions are imposed on the solutions.     

Numerical methods for the QCD overlap operator: III. Nested iterations

The numerical and computational aspects of chiral fermions in lattice quantum chromodynamics are extremely demanding. In the overlap framework, the computation of the fermion propagator leads to a nested iteration where the matrix vector multiplications in each step of an outer iteration have to be accomplished by an inner iteration; the latter approximates the product of the sign function of the hermitian Wilson fermion matrix with a vector.In this paper we investigate aspects of this nested paradigm. We examine several Krylov subspace methods to be used as an outer iteration for both propagator computations and the Hybrid Monte-Carlo scheme. We establish criteria on the accuracy of the inner iteration which allow to preserve an a priori given precision for the overall computation. It will turn out that the accuracy of the sign function can be relaxed as the outer iteration proceeds. Furthermore, we consider preconditioning strategies, where the preconditioner is built upon an inaccurate approximation to the sign function. Relaxation combined with preconditioning allows for considerable savings in computational efforts up to a factor of 4 as our numerical experiments illustrate. We also discuss the possibility of projecting the squared overlap operator into one chiral sector.     

Numerical Approximations for the Cahn–Hilliard Phase Field Model of the Binary Fluid-Surfactant System

In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn–Hilliard equations. The main challenge in solving the system numerically is how to develop easy-to-implement time stepping schemes while preserving the unconditional energy stability. We solve this issue by developing two linear and decoupled, first order and a second order time-stepping schemes using the so-called “invariant energy quadratization” approach for the double well potentials and a subtle explicit-implicit technique for the nonlinear coupling potential. Moreover, the resulting linear system is well-posed and the linear operator is symmetric positive definite. We rigorously prove the first order scheme is unconditionally energy stable. Various numerical simulations are presented to demonstrate the stability and the accuracy thereafter.     

Density-driven flow and solute transport problems. A 2-D numerical model based on the network simulation method

The network simulation method, based on the formal equivalence between physical systems and electrical networks, solves numerical problems of relatively mathematical complexity in a versatile, efficient and computationally fast way. In this paper, the method is applied for the first time to the design of a general purpose model for simulating two-dimensional transient density-driven flow and solute transport through porous media, a mathematical model made up by coupled, nonlinear differential equations. Using the Boussinesq approximation and the stream function formulation, the model is used to solve two typical problems related with groundwater flows. Isochlor concentration and stream function curves are presented and successfully compared with those of other authors. Simulation is carried out using the digital computer program Pspice with relatively low computing times.     

Approximation of the Crank-Nicholson method by the iterated dynamic-theta method

Choptuik's iterated Crank-Nicholson method has become a popular algorithm for solving partial differential equations in computational physics. We generalize Choptuik's explicit iteration approach to implicit finite difference schemes, by the introduction of a novel method with an iteration step dependent parameter and analyze its stability and computational efficiency.     

Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss the most used approaches to the numerical solution of the Sturm-Liouville problem: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initial-value solver; and coefficient approximation methods. Special attention will be paid to techniques that yield uniform approximation over the whole eigenvalue spectrum and that allow large steps even for high eigenvalues.     

A comparable study of image approximations to the reaction field

The recently developed high-order accurate multiple image approximation to the reaction field for a charge inside a dielectric sphere [J. Comput. Phys. 223 (2007) 846-864] is compared favorably to other commonly employed reaction field schemes. These methods are of particular interest because they are useful in the study of biological macromolecules by the Monte Carlo and Molecular Dynamics methods.     

An iterative method to compute the sign function of a non-Hermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential

The overlap Dirac operator in lattice QCD requires the computation of the sign function of a matrix. While this matrix is usually Hermitian, it becomes non-Hermitian in the presence of a quark chemical potential. We show how the action of the sign function of a non-Hermitian matrix on an arbitrary vector can be computed efficiently on large lattices by an iterative method. A Krylov subspace approximation based on the Arnoldi algorithm is described for the evaluation of a generic matrix function. The efficiency of the method is spoiled when the matrix has eigenvalues close to a function discontinuity. This is cured by adding a small number of critical eigenvectors to the Krylov subspace, for which we propose two different deflation schemes. The ensuing modified Arnoldi method is then applied to the sign function, which has a discontinuity along the imaginary axis. The numerical results clearly show the improved efficiency of the method. Our modification is particularly effective when the action of the sign function of the same matrix has to be computed many times on different vectors, e.g., if the overlap Dirac operator is inverted using an iterative method.     

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

In this paper, we consider the numerical approximation of a general second order semilinear stochastic spartial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise with finite trace and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square \(L^2\) norm. Numerical experiments to sustain theoretical results are provided.     

Numerical analysis and physical simulations for the time fractional radial diffusion equation

We do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field. Based on rewriting the equation in a new form, we first present two kinds of implicit finite difference schemes for numerically solving the equation. Then we strictly establish the stability and convergence results. We prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm. Some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes. Finally, we do the physical simulations. Some interesting physical phenomena are revealed; we verify that the long time asymptotic survival probability ∝t^−α, but independent of the dimension, where α is the anomalous diffusion exponent.