Second-order Accurate Finite Difference Scheme Research Articles

Model and numerical method for soliton propagation through thermal medium based on nonlinear Schrödinger and heat transfer equations

We present a mathematical model and numerical method for simulating soliton propagation through thermal medium. The model is developed based on the nonlinear Schrödinger (NLS) equation coupled with a heat transfer equation. The well-posedness of the model and stability of the solution are analyzed. We then develop a second-order accurate finite difference scheme for solving this problem and prove the numerical solution to be bounded in l∞ norm. Furthermore, the scheme is proved to be second-order convergent in l2 norm. Numerical examples are given to verify the model and test the accuracy of the numerical scheme for simulating soliton propagations through the thermal medium.

Conservative second-order accurate finite-difference scheme for the coupled Maxwell-Dirac equations

The recent development of nanoelectronic devices that incorporate Dirac materials has highly increased the need for adequate simulation and modelling tools. This paper introduces an accurate, multiphysics finite-difference time-domain method to solve the pertinent Maxwell-Dirac equations. The stability criterion for the Dirac equation with electromagnetic fields is derived, which reduces to the Courant-Friedrichs-Lewy condition in the absence of electromagnetic fields. Validation examples show the second-order accuracy of the novel fully coupled Maxwell-Dirac scheme and illustrate that total norm and energy are caonserved within a relative error of order 10−4. The method is applied to a ZrTe5 waveguide and it is found that even at low field strengths, the charge carriers can be accelerated to 80% of the Fermi velocity. Furthermore, the flexibility of the advocated method allows for the seamless integration into existing computational electromagnetics frameworks and the possible extension to higher-order schemes.

A Numerical Method for a Heat Conduction Model in a Double-Pane Window

In this article, we propose a one-dimensional heat conduction model for a double-pane window with a temperature-jump boundary condition and a thermal lagging interfacial effect condition between layers. We construct a second-order accurate finite difference scheme to solve the heat conduction problem. The designed scheme is mainly based on approximations satisfying the facts that all inner grid points has second-order temporal and spatial truncation errors, while at the boundary points and at inter-facial points has second-order temporal truncation error and first-order spatial truncation error, respectively. We prove that the finite difference scheme introduced is unconditionally stable, convergent, and has a rate of convergence two in space and time for the L∞-norm. Moreover, we give a numerical example to confirm our theoretical results.

A new absorbing layer for simulation of wave propagation based on a KdV model on unbounded domain

Simulation for wave propagation based on the Korteweg-de Vries (KdV) equation on unbounded domain requires the computational domain to be bounded and hence an absorbing boundary condition is needed so that when the wave arrives the boundary it will not be reflected and destroy the simulation. In this article, we present a new absorbing (or called lossy) layer for simulation of wave propagation based on a KdV model on unbounded domain with non-homogeneous boundary condition at one end, and propose a two-level second-order accurate finite difference scheme for solving this KdV problem. We prove that both the analytical solution and the numerical solution are stable in L2-norm and l2-norm, respectively, and they decay in the lossy layer. Numerical examples are given to illustrate the new method and verify its effectiveness.

Time-dispersion correction for arbitrary even-order Lax-Wendroff methods and the application on full-waveform inversion

A second-order accurate finite-difference (FD) approximation is commonly used to approximate the second-order time derivative of a wave equation. The second-order accurate FD scheme may introduce time dispersion in wavefield extrapolation. Lax-Wendroff methods can suppress such dispersion by replacing the high-order time FD error terms with space FD error-correcting terms. However, the time dispersion cannot be completely eliminated and the computation cost dramatically increases with increasing order of (temporal) accuracy. To mitigate the problem, we have extended the existing time-dispersion correction scheme for the second- or fourth-order Lax-Wendroff method to a scheme for arbitrary even-order methods, which uses the forward and inverse time-dispersion transform (FTDT and ITDT) to add and remove the time dispersion from synthetic data. We test the correction scheme using a homogeneous model and the Sigsbee2A model. The modeling examples suggest that the use of derived FTDT and ITDT pairs on high-order Lax-Wendroff methods can effectively remove time-dispersion errors from high-frequency waves while using longer time steps than allowed in low-order Lax-Wendroff methods. We investigate the influence of the time dispersion on full-waveform inversion (FWI) and show an antidispersion workflow. We apply the FTDT to source terms and recorded traces before inversion, resulting in the source and adjoint wavefields containing equal time dispersion from source-side wave propagation and the modeled and observed traces accumulating equal time dispersion from source- and receiver-side wave propagation. The inversion results reveal that the antidispersion workflow is capable of increasing the accuracy of FWI for arbitrary even-order Lax-Wendroff methods. In addition, the high-order method can obtain better inversion results compared to the second-order method with the same antidispersion workflow.

Difference Scheme for the Numerical Solution of the Burgers Equation

A second-order accurate finite-difference scheme based on existing methods is proposed for the numerical solution of the one-dimensional Burgers equation. A stability condition is given under which the integration time step does not depend on the value of the viscous term. The numerical results produced by the scheme are compared with the exact solution of the Burgers equation.

Numerical Investigation of Unsteady Multiphase Nanofluid-Free Convection Flow About a Vertical Cylinder With Non-Uniform Temperature

Abstract A numerical investigation on the transient-free convection flow of the multiphase nanofluid past a vertical cylinder, which has a power-law variation surface temperature along with the height, is presented. The problem has typical engineering applications involving cooling of vertical cylindrical rods in mechanical/manufacturing systems, cooling of nuclear reactors, and the design of other advanced cooling technologies. Buongiorno’s model is applied in this research, which incorporates thermophoresis and Brownian diffusion effects of nanoparticles. The zero-volume flux condition is implemented for nanoparticle concentration at the boundary to obtain realistic results. A robust second-order accurate finite-difference scheme of Crank–Nicolson type is applied to tackle the system of coupled non-linear partial differential equations numerically. The impacts of time, variable surface temperature power-law exponent, Brownian and thermophoresis parameters are investigated on nanofluid flow and heat transfer aspects. The decisive finding suggests that the effect of the power-law exponent of the variable wall temperature is to reduce the nanoparticle relocation, velocity, and temperature in the nanofluid boundary layer causing the heat transfer enhancement. The skin-friction decreased significantly with the rise of the power-law exponent of the wall temperature. The present numerical scheme is corroborated by comparing the average skin-friction results with the available literature for clear fluid.

A second-order accurate finite-difference scheme for the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation

In this paper, we introduce a wide family of practical numerical methods with two weighing parameters to solve the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation from population dynamics. Our approach is based on the classical Crank-Nicholson method combined with method of lagging to obtain temporally and spatially second-order accurate numerical estimates. Indeed, for a special value of parameters both equal to half we prove that the scheme is second-order accurate in time and in space. In this case, stability, consistency, and (therefore) convergence of the method are examined and it is shown that the method is unconditionally stable under a non-restrictive condition. Two numerical examples are presented and compared with the exact analytical solutions for their order of convergence

A second order unconditionally stable scheme for the modified phase field crystal model with elastic interaction and stochastic noise effect

In this paper, we extend the phase field crystal model to the modified phase field crystal model which includes diffusive dynamics, elastic interaction, and stochastic noises effect. We present a second-order accurate semi-implicit finite difference scheme for the modified phase field crystal model. The resulting scheme is based on the stabilized splitting method and Crank–Nicolson method. The nonlinear term is linearized by the Taylor series. The resulting scheme is linear at each time step, which makes it easy to be implemented and efficient to be solved by using the linear multigrid solver. We prove that the resulting scheme is unconditionally energy stable. Various numerical experiments are conducted to verify the accuracy and efficiency of our proposed algorithm.

Natural convection flow of Rivlin–Ericksen fluid and heat transfer through non-Darcy medium with radiation

The natural convection of non-Newtonian fluids between parallel plates has many engineering applications such as heat exchangers, cooling of electronic equipments, nuclear reactors, solar devices, polymer processing industries, food industries, and petroleum reservoirs. Numerical solution is introduced to solve the governing equations of natural convection of non-Newtonian (Rivlin–Ericksen) fluid flow and heat transfer under the influences of non-Darcy resistance force, constant pressure gradient, dissipation, and radiation. The novelty of this article is to solve this problem between parallel plates channel instead of one plate. The fluid flows between two heated parallel plates that are kept at constant temperatures. Second-order accurate finite difference schemes transform the coupled non-linear differential (momentum and energy) equations to linearized system of algebraic equations. Some comparisons are made to study the convergence and stability of the present results. Effects of parameters of fluid and heat on the velocity field, temperature, skin friction factor, and Nusselt number are illustrated and discussed. The present results and their comparisons with available results are listed and shown in tables and figures. The present results show that the numerical solution is of good agreement with previous analytical and numerical solutions.

Magnetohydrodynamic flow of continuous dusty particles and non-Newtonian Darcy fluids between parallel plates

The coupled unsteady power-law conducting fluid flow and continuous dusty viscous fluid flow under the influence of magnetic field are solved using the finite difference method. The resistive forces of Darcy porous medium and the external uniform magnetic field are applied on the flow. The second-order accurate finite difference schemes are applied on the coupled governing equations to transform the non-linear partial differential equations to linearized system of algebraic equations. This system is solved iteratively using the generalized Thomas algorithm. Some results are introduced to study the convergence and stability of the present works. The effects of non-Newtonian fluid, continuous dusty particles, Darcy model on the velocity field, and friction factor of both the fluid and dust particles phases are demonstrated.

Second-Order Accurate Finite-Difference Scheme for Solving the Problem of Elastic Wave Diffraction by the Anisotropic Gradient Layer

The boundary value problem for the Lame equations for the problem of elastic wave diffraction by an anisotropic layer with continuously varying elastic parameters is considered. The original problem is reduced to the boundary value problem for a system of ordinary differential equations of the given form. The finite-difference scheme is obtained by the method of approximation of integral identities. The theorem is proved that the error of approximation of the solution has a second order of accuracy for sufficiently continuous values of the elements of the elasticity tensor. Numerical results confirming theoretical conclusions are given.

A fourth-order accurate quasi-variable mesh compact finite-difference scheme for two-space dimensional convection-diffusion problems

We discuss a new nine-point fourth-order and five-point second-order accurate finite-difference scheme for the numerical solution of two-space dimensional convection-diffusion problems. The compact operators are defined on a quasi-variable mesh network with the same order and accuracy as obtained by the central difference and averaging operators on uniform meshes. Subsequently, a high-order difference scheme is developed to get the numerical accuracy of order four on quasi-variable meshes as well as on uniform meshes. The error analysis of the fourth-order compact scheme is described in detail by means of matrix analysis. Some examples related with convection-diffusion equations are provided to present performance and robustness of the proposed scheme.

StaRMAP---A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent PN and SPN equations, of radiative transfer. The method, which works for arbitrary moment order N , makes use of the specific coupling between the moments in the PN equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.

Diagnostic Computation of Moisture Budgets in the ERA-Interim Reanalysis with Reference to Analysis of CMIP-Archived Atmospheric Model Data*

AbstractThe diagnostic evaluation of moisture budgets in archived atmosphere model data is examined. Sources of error in diagnostic computation can arise from the use of numerical methods different from those used in the atmosphere model, the time and vertical resolution of the archived data, and data availability. These sources of error are assessed using the climatological moisture balance in the European Centre for Medium-Range Weather Forecasts Interim Re-Analysis (ERA-Interim) that archives vertically integrated moisture fluxes and convergence. The largest single source of error arises from the diagnostic evaluation of divergence. The chosen second-order accurate centered finite difference scheme applied to the actual vertically integrated moisture fluxes leads to significant differences from the ERA-Interim reported moisture convergence. Using daily data, instead of 6-hourly data, leads to an underestimation of the patterns of moisture divergence and convergence by midlatitude transient eddies. A larger and more widespread error occurs when the vertical resolution of the model data is reduced to the 8 levels that is quite common for daily data archived for the Coupled Model Intercomparison Project (CMIP). Dividing moisture divergence into components due to the divergent flow and advection requires bringing the divergence operator inside the vertical integral, which introduces a surface term for which a means of accurate evaluation is developed. The analysis of errors is extended to the case of the spring 1993 Mississippi valley floods, the causes of which are discussed. For future archiving of data (e.g., by CMIP), it is recommended that monthly means of time-step-resolution flow–humidity covariances be archived at high vertical resolution.

Seismic wavefield modeling in media with fluid-filled fractures and surface topography

We present a finite difference (FD) method for the simulation of seismic wave fields in fractured medium with an irregular (non-flat) free surface which is beneficial for interpreting exploration data acquired in mountainous regions. Fractures are introduced through the Coates-Schoenberg approach into the FD scheme which leads to local anisotropic properties of the media where fractures are embedded. To implement surface topography, we take advantage of the boundary-conforming grid and map a rectangular grid onto a curved one. We use a stable and explicit second-order accurate finite difference scheme to discretize the elastic wave equations (in a curvilinear coordinate system) in a 2D heterogeneous transversely isotropic medium with a horizontal axis of symmetry (HTI). Efficiency tests performed by different numerical experiments clearly illustrate the influence of an irregular free surface on seismic wave propagation in fractured media which may be significant to mountain seismic exploration. The tests also illustrate that the scattered waves induced by the tips of the fracture are re-scattered by the features of the free surface topography. The scattered waves provoked by the topography are re-scattered by the fractures, especially Rayleigh wave scattering whose amplitudes are much larger than others and making it very difficult to identify effective information from the fractures.

Gravity-driven azimuthal flow of a layer of thixotropic fluid on the inner surface of a horizontal tube

The gravity-driven azimuthal flow of a layer of thixotropic paint on the inner surface of a horizontal tube is studied, considering surface tension effects. Using the lubrication theory, it was shown that a non-linear, fourth-order partial differential equation governs the time evolution of the paint layer thickness distribution along the azimuthal coordinate. Three parameters arise in the analysis, namely, the Bond number and two rheology-related parameters. The governing equation is integrated via a second-order accurate finite-difference scheme. The results showed that, in situations where the capillary force dominates ( Bo < 1), displacement of the paint after application is very slow. For situations where the gravity force dominates ( Bo > 1), an undulation on the interface arises near the tube bottom at sufficiently large times.

Three-Dimensional Wave-Field Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface

Modeling of seismic-wave propagation in anisotropic medium with irregular topography is beneficial to interpret seismic data acquired by active and pas- sive source seismology conducted in areas of interest such as mountain ranges and basins. The major challenge in this context is the difficulty in tackling the irregular free-surface boundary condition in a Cartesian coordinate system. To implement sur- facetopography,weusetheboundary-conforminggridandmaparectangulargridonto a curved grid. We use a stable and explicit second-order accurate finite-difference scheme to discretize the elastic wave equations (in a curvilinear coordinate system) in a 3D heterogeneous transversely isotropic medium. The free-surface boundary con- ditions are accurately applied by introducing a discretization that uses boundary- modified difference operators for the mixed derivatives in the governing equations. The accuracy of the proposed method is checked by comparing the numerical results obtained by the trial algorithm with the analytical solutions of the Lamb's problem, for anisotropicmediumandatransverselyisotropicmediumwithaverticalsymmetryaxis, respectively. Efficiency tests performed by different numerical experiments illustrate clearlytheinfluenceofanirregular(nonflat)freesurfaceonseismic-wavepropagation.

Accurate finite difference schemes for solving a 3D micro heat transfer model in an N-carrier system with the Neumann boundary condition in spherical coordinates

In this study, we propose a 3D generalized micro heat transfer model in an N-carrier system with the Neumann boundary condition in spherical coordinates, which can be applied to describe the non-equilibrium heating in biological cells. Two improved unconditionally stable Crank–Nicholson schemes are then presented for solving the generalized model. In particular, we delicately adjust the location of the interior grid point that is next to the boundary so that the Neumann boundary condition can be applied directly without discretization. As such, a second-order accurate finite difference scheme without using any fictitious grid points is obtained. The convergence rates of the numerical solution are tested by an example. Results show that the convergence rates of the present schemes are about 2.0 with respect to the spatial variable r , which improves the accuracy of the Crank–Nicholson scheme coupled with the conventional first-order approximation for the Neumann boundary condition.

Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems

We introduce a class of stable and second-order accurate finite difference schemes that resolve a number of problems when approximating the solution of option pricing models in computational finance using the finite difference method (FDM). In particular, we show how to avoid having to apply truncation methods to the domain of integration as well as the resulting ad-hoc experimentation in choosing suitable numerical boundary conditions that we must apply on the boundary of the truncated domain. Instead, we transform the PDE option model (which is originally defiined on a semi-infinite interval) to a PDE that is defined on a bounded interval. We then apply the elegant Fichera theory to help us determine which boundary conditions to apply to the transformed PDE. We show that the new system is well-posed by proving energy inequalities in the space of square-integrable functions. Having done that, we adapt the Alternating Direction Explicit (ADE) (a method that dates from the first half of the last century) method to compute an approximation to the solution of the transformed PDE. We prove that the explicit scheme is unconditionally stable and second-order accurate. The scheme is very easy to model, to program and to parallelise and the generalization to n-factor problems is easily motivated. We examine equity problems and we compare the ADE approach (in terms of accuracy and speedup) with more traditional finite difference schemes and with the Monte Carlo method. Finally, we stress-test the scheme by varying critical parameters over a spectrum of values and the results are noted. The methods in this article - in particular the combination of pure and applied techniques - are relatively new in the computational finance literature in our opinion, in particular the realization that a PDE on a semi-infinite interval can be transformed to one on a bounded interval and that the new PDE can be approximated by schemes other than Crank-Nicolson and its workarounds. Finally, we shall report on n-factor models in future articles.