Order Accurate Finite Difference Scheme Research Articles

A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation

We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (μ(x)u x ) x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (μ(x)u x ) x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.

Moment preserving schemes for Euler equations

A high order accurate finite difference scheme is proposed for one-dimensional Euler equations. In the scheme a set of first three moments of each signal are preserved during the updating. The scheme is one of 5th order in space and 4th order in time. This feature is different from that in typical existing methods in which the use of the first three polynomials results in only 3rd order accuracy in space. The scheme has different features from the existing high order schemes, and the most noticeable are the simultaneous discretization both in space and time, and the use of moments of Riemann invariants instead of primitive physical variables. Numerical examples are given to show the accuracy of the scheme and its robustness for the flows involving shocks.

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.

Fully and Semi-discrete Fourth-order Schemes for Hyperbolic Conservation Laws

AbstractA new fourth order accurate centered finite difference scheme for the solution of hyperbolic conservation laws is presented. A technique of making the fourth order scheme TVD is presented. The resulting scheme can avoid spurious oscillations and preserve fourth order accuracy in smooth parts. We discuss the extension of the TVD scheme to the nonlinear scalar hyperbolic conservation laws. For nonlinear systems, the TVD constraint is applied by solving shallow water equations. Then, we propose to use this fourth order flux as a building block in spatially fifth order weighted essentially non-oscillatory (WENO) schemes. The numerical solution is advanced in time by the third order TVD Runge — Kutta method. The performance of the scheme is assessed by solving test problems. The numerical results are presented and compared to the exact solutions and other methods.

A Second Order Accurate Finite Difference Scheme for the Heat Equation on Irregular Domains and Adaptive Grids

AbstractWe present a finite difference scheme for solving the variable coefficient heat equations with Dirichlet boundary conditions on irregular domains. A quadtree data structure is used to represent the non-graded adaptive Cartesian grids, and the interface is represented by the zero value points of the level set function. Numerical results in two spatial dimensions demonstrate second order accuracy for both the solution and its gradient.

Domain decomposition technique for aeroacoustic simulations

A computational tool for the simulation of time harmonic sound propagation in open domains is proposed. It consists of a two step technique. In the first step the viscous flow field is calculated, solving the incompressible Navier–Stokes equations. In the second step the acoustic field is obtained solving the Lighthill's wave equation. The two calculations are performed on two different grids, both formed by non-conforming subdomains. The Lighthill's wave equation is solved in the frequency domain, with a high order accurate compact finite difference scheme. To test the acoustic solver, the phenomenon of wave diffraction past an obstacle is calculated. The results confirm that the interface transmission conditions as well the non-reflecting boundary conditions do not affect the accuracy of the scheme. The complete aeroacoustic technique is applied to the simulation of the noise radiation from a cavity with a grazing subsonic laminar boundary layer. The flow field presents a periodic vortex shedding past the cavity, the vortical flow being a noise source. The numerical results show the existence of the directivity character of the acoustic radiation.

Numerical simulation of Richtmyer-Meshkov instability

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1:20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.

Analysis and simulation for a system of chemical reaction equations with a vortex formulation

This work presents analytical and numerical results for the two dimensional molecular mixing and chemical reaction processes using a vortex formulation. The particular model studied here is the single-step, irreversible, exothermic Arrhenius type reaction c A + c B → c P , in an incompressible fluid under Neumann boundary conditions. We establish the existence of a unique classical solution considering the weak coupling of the reactants via advection by a independently developing velocity, and we prove the existence and uniqueness of the solution via a semigroup formulation and the maximum principle. For the numerical solution a second order accurate spatial finite difference scheme is used with a second order accurate Runge–Kutta method for the time step. The behaviour of the concentrations c A and c B , the temperature, the reaction rate, and product concentration c P , are obtained over a wide range of Reynolds and Damköhler numbers.

Numerical simulation of forced convection flow past a parabolic cylinder embedded in porous media

Numerical solutions are presented for steady two‐dimensional symmetric flow past a parabolic cylinder embedded in porous media. For this study, the full Navier–Stokes equations (combined with the Brinkman–Forchheimer‐extended Darcy model) and energy equation in parabolic coordinates were solved. A second order accurate finite difference scheme on a non‐uniform grid was used. A wide range of Reynolds number (Re) is studied for different values of Prandtl number (Pr). It is found that the pressure, skin friction and Nusselt number decreases as the Darcy number (Da) decreases and/or the Inertia parameter (Λ) increases.

Numerical study on Landau damping

We present a numerical study of the so-called Landau damping phenomenon in the Vlasov theory for spatially periodic plasmas in a nonlinear setting. It shows that the electric field does decay exponentially to zero as time goes to infinity with general analytical initial data which are close to a Maxwellian. The time decay depends on the length of the period as well as the closeness between the initial data and the Maxwellian. A similar pattern is observed if the Maxwellian is replaced by other algebraically decaying homogeneous equilibria with a single maximum, or even by some homogeneous equilibria with small double-humps. The numerical method used is a high order accurate hybrid spectral and finite difference scheme which is carefully calibrated with the well-known decay theory for the corresponding linear case, to guarantee a reliable resolution free of numerical artifacts for a long time integration.

Hydrodynamic and heat transfer characteristics of laminar flow past a parabolic cylinder with constant heat flux

Steady, two-dimensional, symmetric, laminar and incompressible flow past parabolic bodies in a uniform stream with constant heat flux is investigated numerically. The full Navier–Stokes and energy equations in parabolic coordinates with stream function, vorticity and temperature as dependent variables were solved. These equations were solved using a second order accurate finite difference scheme on a non-uniform grid. The leading edge region was part of the solution domain. Wide range of Reynolds number (based on the nose radius of curvature) was covered for different values of Prandtl number. The flow past a semi-infinite flat plate was obtained when Reynolds number is set equal to zero. Results are presented for pressure and temperature distributions. Also local and average skin friction and Nusselt number distributions are presented. The effect of both Reynolds number and Prandtl number on the local and average Nusselt number is also presented.

Numerical solutions of the Navier‐Stokes and energy equations for laminar incompressible flow past parabolic bodies

Numerical solutions are presented for steady two‐dimensional symmetric flow past parabolic bodies in a uniform stream parallel to its axis. For this study, the full Navier‐Stokes equations and energy equation in parabolic coordinates were solved. A second order accurate finite difference scheme on a non‐uniform grid was used. The solution domain does not exclude the leading edge region as it is usually done with boundary layer flows. A wide range of Reynolds number (Re) is studied for different values of Prandtl number (Pr). It is found that the average Nusselt number (Nu) increases as (Pr) increases meanwhile, (Nu) decreases with the increase in (Re).

Flux limiters fourth and sixth order high resolution method for hyperbolic conservation laws

This paper presents new fourth and six order accurate explicit finite difference scheme for hyperbolic conservation laws. The technique of obtaining high resolution, total variation non increasing oscillation free and explicit scalar difference scheme is derived by adding multi-limiters of antidiffusive flux to a first order scheme.

Preconditioning by approximations of the discrete Laplacian for 2‐D non‐linear free convection elliptic equations

The paper analyses the preconditioning of non‐linear nonsymmetric equations with approximations of the discrete Laplace operator. The test problems are non‐linear 2‐D elliptic equations that describe natural convection, Darcy flow, in a porous medium. The standard second order accurate finite difference scheme is used to discretize the models’ equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the preconditioned generalised conjugate gradient (PGCG) methods as inner iteration. Three PGCG algorithms, CGN, CGS and GMRES, are tested. The preconditioning with discrete Laplace operator approximations consists of replacing the solving of the equation with the preconditioner by a few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky (IC) and multigrid (MG). The numerical results show that MG preconditioning leads to mesh independence. CGS is the most robust algorithm but its efficiency is lower than that of GMRES.

Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow

Conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. Existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out. In particular, it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass, momentum, and kinetic energy. This deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems. Finite difference schemes in a collocated grid system are also analyzed, and a violation of kinetic energy conservation is revealed. The predicted conservation properties are demonstrated numerically in simulations of inviscid white noise, performed in a two-dimensional periodic domain. The proposed fourth order schemes in a staggered grid system are generalized for the case of a non-uniform mesh, and the resulting scheme is used to perform large eddy simulations of turbulent channel flow.

Numerical Simulations for Radiation Hydrodynamics. I. Diffusion Limit

A second order accurate finite difference scheme is proposed for multidimensional radiation hydrodynamical equations in a diffusion limit. The radiation hydrodynamical equations in the limit constitute a hyperbolic system of conservation laws plus radiative heat transfer. A Godunov scheme including linear and nonlinear Riemann solvers is proposed for the set of conservation laws. The scheme with the linear Riemann solver works well for relatively strong shocks. The nonlinear Riemann solver is specially designed for flows involving strong shocks. The radiative heat conduction is treated implicitly. The treatment possesses a number of advantages over typical implicit methods. The most notable are the second order accuracy in both space and time, quick damping of numerical errors when the size of time steps is large, iterative solver and the fast convergence, the accurate treatment for the nonlinearity, and the energy conservation. Numerical examples are given to show the features of the schemes.

High Resolution Schemes for Hyperbolic Conservation Laws

A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.

Preconditioned cg‐like methods for solving non‐linear convection—diffusion equations

The paper presents the numerical performance of the preconditioned generalized conjugate gradient (PGCG) methods in solving non‐linear convection — diffusion equations. Three non‐linear systems which describe a non‐isothermal chemical reactor, the chemically driven convection in a porous medium and the incompressible steady flow past a sphere are the test problems. The standard second order accurate centred finite difference scheme is used to discretize the models equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the PGCG algorithm as inner iteration. Three PGCG techniques, which emerge to be the best performing, are tested. Laplace‐type operators are employed for preconditioning. The results show that the convergence of the PGCG methods depends strongly on the convection—diffusion ratio. The most robust algorithm is GMRES. But even with GMRES non‐convergence occurs when the convection—diffusion ratio exceeds a limit value. Thi...

Accurate numerical solution of compressible, linear stability equations

A compact fourth order accurate finite difference scheme is proposed for solving the eigenvalue problem of the temporal and spatial stability of three-dimensional compressible boundary layers. It is applied to the stability study of the flow on a laminar flow control swept wing. The results indicate that the present method provides an accurate and efficient way of calculating the eigenvalues and eigensolutions of the compressible linear stability equations.

Numerical and experimental analysis of unsteady laminar forced convection in channels

Unsteady heat transfer for fully developed laminar air flow with a parabolic velocity profile through a parallel plate channel, subjected to sinusoidally varying inlet temperature is considered. Numerical as well as experimental results are reported. A second order accurate explicit finite difference scheme is used in the numerical solution to the energy equation. A general boundary condition of fifth kind which accounts for both external convection and wall thermal capacitance effects is introduced. Experiments are performed on an apparatus specially designed for the investigation of unsteady forced convection heat transfer, resulting from a sinusoidal variation of the inlet temperature in rectangular channels. The variations in the amplitude of periodic temperature oscillations along the centerline of the rectangular channel, obtained from the numerical model and experiments are compared on both a quantitative and qualitative basis.