Semi-discrete Difference Schemes Research Articles

Erratum to: Semi-Implicit and Semidiscrete Difference Schemes for Solving a Nonstationary Kinetic Equation of Thermal Radiative Transfer and Energy Equation

Erratum to: Semi-Implicit and Semidiscrete Difference Schemes for Solving a Nonstationary Kinetic Equation of Thermal Radiative Transfer and Energy Equation

Semi-Implicit and Semidiscrete Difference Schemes for Solving a Nonstationary Kinetic Equation of Thermal Radiative Transfer and Energy Equation

Semi-Implicit and Semidiscrete Difference Schemes for Solving a Nonstationary Kinetic Equation of Thermal Radiative Transfer and Energy Equation

Finite Difference Methods for Caputo–Hadamard Fractional Differential Equations

In this paper, we study finite difference methods for fractional differential equations (FDEs) with Caputo–Hadamard derivatives. First, smoothness properties of the solution are investigated. The fractional rectangular, $${L}_{\mathrm{log},1}$$ interpolation, and modified predictor–corrector methods for Caputo–Hadamard fractional ordinary differential equations (FODEs) are proposed through approximating the corresponding equivalent Volterra integral equations. The stability and error estimate of the derived methods are proved as well. Then, we investigate finite difference methods for fractional partial differential equations (FPDEs) with Caputo–Hadamard derivative. By applying the constructed L1 scheme for approximating the time fractional derivative, a semi-discrete difference scheme is derived. The stability and convergence analysis are shown too in detail. Furthermore, a fully discrete scheme is established by the standard second-order difference scheme in spacial direction. Stability and error estimate are also presented. The numerical experiments are displayed to verify the theoretical results.

A difference method for the McKean–Vlasov equation

We analyze a model equation arising in option pricing. This model equation takes the form of a nonlinear, nonlocal diffusion equation. We prove the well posedness of the Cauchy problem for this equation. Furthermore, we introduce a semidiscrete difference scheme and show its rate of convergence.

Thermal stability and entropy generation of unsteady reactive hydromagnetic Powell-Eyring fluid with variable electrical and thermal conductivities

Theoretical analysis of thermal stability and inherent irreversibility of reactive hydromagnetic Powell-Eyring fluid flow in a fixed vertical channel with variable electrical and thermal conductivities is examined. The conducting reactive fluid is influenced by gravity and axial pressure gradient under bimolecular kinetic rate law. Ignoring the material assumptions, the governing dimensionless equations are solved by finite semi-discretization difference scheme along with integration Fehlberg Runge-Kutta method. The results for the momentum and heat equations, Bejan number, thermal runaway, entropy generation and current density are graphically presented under different thermophysical parameters. The computational results for the shear stress and heat transfer at the wall is also presented and discussed quantitatively. It was observed that the parameter which enhance the rate of entropy generation also augment Bejan number with large pitch ratio. Hence, entropy generation can be minimized at low dissipation and material variables.

A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.

Nonlinear standing waves in 2-D acoustic resonators

This paper deals with 2-D simulation of finite-amplitude standing waves behavior in rectangular acoustic resonators. Set of three partial differential equations in third approximation formulated in conservative form is derived from fundamental equations of gas dynamics. These equations form a closed set for two components of acoustic velocity vector and density, the equations account for external driving force, gas dynamic nonlinearities and thermoviscous dissipation. Pressure is obtained from solution of the set by means of an analytical formula. The equations are formulated in the Cartesian coordinate system. The model equations set is solved numerically in time domain using a central semi-discrete difference scheme developed for integration of sets of convection–diffusion equations with two or more spatial coordinates. Numerical results show various patterns of acoustic field in resonators driven using vibrating piston with spatial distribution of velocity. Excitation of lateral shock-wave mode is observed when resonant conditions are fulfilled for longitudinal as well as for transversal direction along the resonator cavity.

Central difference regularization method for the Cauchy problem of the Laplace’s equation

In this paper, we consider the Cauchy problem for the Laplace’s equation where the Cauchy data is given at x = 0 and the solution is sought in the interval 0 < x < 1. A semi-discrete difference schemes together with a choice of regularization parameter is presented and error estimate is obtained. A numerical example shows that the method works effectively.

High Order Symplectic Schemes for the Sine-Gordon Equation*

In this paper, taking the sine-Gordon equation as an example, we present a new method to construct the symplectic schemes for Hamilton PDEs. Different from the previous symplectic methods dealing with PDEs, our method is that to view the PDEs as a Hamilton system in Banach space, then to apply the generating functions method to the Hamilton system. After overcoming successfully the essential difficulties on the calculations of high order variation derivatives, we get the semi-discrete difference schemes for the PDEs with arbitrary order of accuracy in time direction. Furthermore the corresponding modified equations of the infinite dimensional Hamiltonian system are obtained from the semi-discretization. We use the central difference operators to discretize the derivatives in space. Thus the resulting full discrete symplectic schemes can be of any order accuracy. Numerical results on collisions of solitons are also presented to show the effectiveness of the schemes.

Statistical Mechanics for Truncations of the Burgers-Hopf Equation: A Model for Intrinsic Stochastic Behavior with Scaling

In this paper we consider both analytically and numerically several finite-dimensional approximations for the inviscid Burgers-Hopf equation. Fourier Galerkin truncation is introduced and studied as a simple one-dimensional model with intrinsic chaos and a well-defined mathematical structure allowing for an equilibrium statistical mechan- ics formalism. A simple scaling theory for correlations is developed that is supported strongly by the numerical evidence. Several semi-discrete difference schemes with similar mathematical properties conserving dis- crete momentum and energy are also considered. The mathematical properties of the difference schemes are analyzed and the behavior of the difference schemes is compared and contrasted with the Fourier Galerkin truncation. Numerical simulations are presented which show similarities and subtle differences between different finite-dimensional approximations both in the deterministic and stochastic regimes with many degrees of freedom.

Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data

We prove the global existence of weak solutions for the Cauchy problem for the Navier-Stokes equations for one-dimensional, isentropic flow when the initial velocity is in L2 and the initial density is in L2 ∩ BV. Solutions are obtained as limits of approximations obtained by building heuristic jump conditions into a semi-discrete difference scheme. This allows for a rather simple analysis in which pointwise control is achieved through piecewise H1 and total variation estimates.

Convergence of semi-discrete difference schemes of abstract Cauchy problems

Convergence of semi-discrete difference schemes of abstract Cauchy problems