Hybrid Difference Methods for the Initial Boundary-Value Problem for Hyperbolic Equations

Centered difference approximations of fourth order in space and second order in time are applied to the mixed initial boundary-value problem for the hyperbolic equation ${u_t} = - c{u_x}$. A method utilizing third order uncentered differences at the boundaries is shown to be stable and to retain an overall fourth order convergence estimate. Several computational examples illustrate the success of these methods for problems with one and two spacial dimensions. Further examples illustrate the effects of approximations of various orders of accuracy used at the boundaries.

Computation of steady and oscillatory convection in saturated porous media

A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

Multidimensional difference schemes with fourth-order accuracy

This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection–diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter-uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.

A nonlinear discontinuity-preserving method for noise removal for band pass signals such as signals modulated with Binary Phase-Shift Keying (BPSK) modulation is proposed in this paper. This method is inspired by the anisotropic diffusion algorithm to remove noise and preserve discontinuities in band pass signals modulated with a single frequency. It is demonstrated here that nonlinear noise removal method for a real valued band pass signal requires a solution for a nonlinear partial differential equation which is of fourth order in space and second order in time. The results presented in this work show better performance in nonlinear noise removal for real valued band pass signals in comparison with the previous work in the literature.

Optimal error estimate of a compact scheme for nonlinear Schrödinger equation

A second-order in time and fourth-order in space, alternating direction implicit finite difference time domain (ADI-FDTD) scheme is presented for radio-wave propagation in a isotropic collisionless plasma. At the same time, the unconditional stability of the higher order ADI-FDTD formulation for plasma is analytically proved. By analysis of the growth factor, the numerical dispersion relation is derived. In addition, the numerical dispersion errors are investigated. From the error analysis and phase velocity data, it is argued that the proposed approach will significantly minimize the dispersion errors compared to the ADI-FDTD method.

This study uses the hybrid finite-difference method (HFDM), which combines the finite-difference and spectral approaches, to describe the Magnetohydrodynamic (MHD) turbulence decay. By using a finite-difference approach in conjunction with a cyclic Penta-diagonal matrix, the numerical algorithm of the hybrid method solves the Navier-Stokes equations and the magnetic field equation with the fourth order's precision in space and second order in time. The spectral approach is used to solve the pressure Poisson equation. The time-dependent turbulence features of this flow were in excellent agreement with the appropriate analytical solution, which is valid for short timeframes, for the classical issue of the 3-D Taylor and Green vortex flow without taking the magnetic field into account. We also show how the effective numerical approach may be utilised to model the decline of magnetohydrodynamic turbulence at various magnetic Reynolds numbers.

A new formulation is presented for the problem of the time-dependent perturbation of bound states. It is based on the hydrodynamic analogy to quantum mechanics and leads to the physical picture of a continuum of particles which oscillate in the vicinity of their equilibrium positions under the influence of the perturbing field. The corresponding mathematical formulation involves a single partial differential equation which is second order in time and fourth order in the spatial coordinates. The procedure is applied to the time-dependent polarizability of the hydrogen atom in its ground state. It was possible to obtain an exact series solution which yields results in full agreement with those previously obtained by other methods.

We investigate the three-dimensional mathematical model of crystal growth called PFC (Phase Field Crystal) in a hyperbolic modification. This model is also called the modified model PFC (originally PFC model is formulated in parabolic form) and allows to describe both slow and rapid crystallization processes on atomic length scales and on diffusive time scales. Modified PFC model is described by the differential equation in partial derivatives of the sixth order in space and second order in time. The solution of this equation is possible only by numerical methods. Previously, authors created the software package for the solution of the Phase Field Crystal problem, based on the method of isogeometric analysis (IGA) and PetIGA program library. During further investigation it was found that the quality of the solution can strongly depends on the discretization parameters of a numerical method. In this report, we show the features that should be taken into account during constructing the computational grid for the numerical simulation.

Based on the method of characteristics, a numerical algorithm for the solution of the Vlasov-Poisson system of equations, in the magnetized plasma with immobile ions is presented. The simulation scheme pushes the phase point along their trajectories while distribution function is fixed [1]. The scheme provides enhanced second order accuracy in time and fourth order in space. Landau-Bernstein paradox has been examined and physical expectations have been achieved qualitatively.

Two hydro turbine fluid flow problems are solved numerically in the paper. Both problems are solved in frames of Euler model using CFD code developed by the authors. The code is based on finite volume artificial compressibility approach. The accuracy of numerical scheme is second order in time and third order in space. For the solution of unsteady problems dual-time-stepping algorithm is used. Complex geometry of a turbine passage is handled using domain decomposition. The first problem considered is a simulation of precessing vortex rope downstream the turbine runner. Numerical results showed that this phenomenon can be captured by Euler model. The second problem is an automatic design (optimization) of runner blade shape with help of Breeder genetic algorithm. Different formulations of the objective function are considered and their influence on the blade shape is demonstrated.

We construct a posteriori error estimators for approximate solutions of linear parabolic equations. We consider discretizations of the problem by modified discontinuous Galerkin schemes in time and continuous Galerkin methods in space. Especially, finite element spaces are permitted to change at different time levels. Exploiting Crank-Nicolson reconstruction idea introduced by Akrivis, Makridakis & Nochetto [2], we derive space-time a posteriori error estimators of second order in time for the Crank-Nicolson-Galerkin finite element method.

In this article, a uniformly convergent numerical scheme is developed to solve a singularly perturbed convection‐diffusion equation with a small delay having a boundary layer along the left side. A priori bounds of continuous solution and its derivatives are discussed. To solve the problem, the Crank–Nicolson scheme in the time direction and the exponentially fitted finite difference scheme in the space direction are used. The stability of the method is analyzed. It is proved that the developed scheme converges uniformly with first order in space and second order in time. To validate the applicability of the theoretical finding of the developed scheme, numerical experiments are carried out by considering two test examples.