High-order Finite Difference Scheme Research Articles

Effects of Wall Temperature on the Evolution of a Large-Amplitude Entropy Wave

AbstractA high-order finite-difference scheme is employed to discretize the complete Navier-Stokes equations in order to simulate the hypersonic unsteady flowfield over a blunt cone. The hypersonic...

Robust hybrid schemes of higher order for singularly perturbed convection-diffusion problems

Abstract A class of linear singularly perturbed convection-diffusion problems in one dimension is discretized on the Shishkin mesh using hybrid higher-order finite-difference schemes. Under appropriate conditions, pointwise convergence uniform in the perturbation parameter e is proved for one of the discretizations. This is done by the preconditioning approach, which enables the proof of e-uniform stability and e-uniform consistency, both in the maximum norm. The order of convergence is almost 3 when e is sufficiently small.

Acceleration of high-order combined compact finite difference scheme for simulating three-dimensional flow and heat transfer problems in GPUs

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

A new eighth-order implicit finite difference method to solve the three-dimensional Helmholtz equation

This paper introduces a novel method for deducing high-order finite difference schemes for the three-dimensional Helmholtz equation, Δp±λ2p=f. The new method can reach an eighth-order of accuracy. It is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives. The scheme is obtained from Taylor series expansion and plane wave theory analysis, and it is constructed from a few modifications to the standard centered finite differences. A classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given. Numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of λ-values. Moreover, the numerical method is capable to solve with great precision the monotone and oscillatory Helmholtz problem with variable λ-values. Thus, the proposed formulation results in an attractive method, easy to implement, with high order of accuracy but nearly the same computational cost as those of an explicit formulation.

High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

This paper develops the high-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order schemes. Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

A New Code for Relativistic Hydrodynamics and its Application to FR II Radio Jets

AbstractTo study the dynamics of relativistic flows in astrophysical objects such as radio jets, we have developed a new special relativistic hydrodynamic (RHD) code based on the weighted essentially non-oscillatory (WENO) scheme, a high-order finite difference scheme. The code includes different WENO versions, and high-order time integration methods such as the 4th-order accurate Runge-Kutta (RK4) and strong stability preserving RK (SSPRK), as well as the equations of state (EOSs) that closely approximate the EOS of the single-component perfect gas in relativistic regime. Additionally, it is optimized for the reproduction of complex structures in multi-dimensional flows, and implements a modification of eigenvalues for the acoustic modes to effectively control carbuncle instability. As the first application of the code, we have simulated ultra-relativistic jets of FR II radio galaxies, and studied the nonlinear flow structures, such as shocks, velocity shear, and turbulence, through large-scale.

Large eddy simulation of propane combustion in a planar trapped vortex combustor

Propane combustion in a trapped vortex combustor (TVC) is characterized via large eddy simulation coupled with filtered mass density function. A computational algorithm based on high order finite difference (FD) schemes, is employed to solve the Eulerian filtered compressible Navier-Stokes equations. In contrast, a Lagrangian Monte-Carlo solver based on the filtered mass density function is invoked to describe the scalar field. The impact of injection strategy on temperature distribution and flame structure in a planar single-cavity TVC is investigated. A fuel jet and an air jet are injected directly into the cavity from the forebody and the afterbody, respectively. Different injection schemes are contemplated by altering fuel and air jet locations representing the different flow and flame structures. The temperature distribution, along with cross-sectional averaged temperature and flame structure, are compared for fuel/air injection strategies. The temperature field reveals that configurations in which both air and fuel jets are located at the cavity-walls midpoint or adjacent to the cavity inferior wall, lead to a more uniform temperature distribution and lower maximum temperature with the latter configuration performing slightly better. While, the former configuration provides the closest cross-sectional averaged temperature to the adiabatic flame temperature. The reaction rate distributions show that the configurations mentioned above lead to a more contained flame, chiefly due to more efficient fuel-air mixing at lower regions of the cavity.

High-order compact scheme for the two-dimensional fractional Rayleigh\u2013Stokes problem for a heated generalized second-grade fluid

In this article, an unconditionally stable compact high-order iterative finite difference scheme is developed on solving the two-dimensional fractional Rayleigh–Stokes equation. A relationship between the Riemann–Liouville (R–L) and Grunwald–Letnikov (G–L) fractional derivatives is used for the time-fractional derivative, and a fourth-order compact Crank–Nicolson approximation is applied for the space derivative to produce a high-order compact scheme. The stability and convergence for the proposed method will be proven; the proposed method will be shown to have the order of convergence O(tau + h^{4}). Finally, numerical examples are provided to show the high accuracy solutions of the proposed scheme.

Time-domain simulation of wave propagation across resonant meta-interfaces

Meta-interface models stem from the homogenization, in a low-frequency dynamic regime, of thin heterogeneous layers that are structured to achieve uncommon properties at the macroscopic level. When the layer is composed of a thin periodic array of highly-contrasted inclusions embedded within a homogeneous background medium then the corresponding effective interface model is characterized by jump conditions that, in the harmonic regime, involve some singular frequency-dependent terms. In this context, the article is concerned with the simulation of transient waves across such resonant meta-interfaces and a numerical method is proposed to handle the associated resonant jump conditions. To do so, a set of auxiliary variables is introduced locally along the interface and an augmented system of first-order equations in time accompanied with local-in-time jump conditions is derived. This system is then discretized on a Cartesian grid and solved using a high-order finite-difference scheme while the complexity associated with the geometry of the interface and the jump conditions is handled using an immersed interface method. A set of numerical examples in 1D and 2D is proposed to illustrate and validate the overall numerical approach, and quantitative comparisons with semi-analytical solutions are also provided.

Multigrid method with eighth-order compact finite difference scheme for Helmholtz equation

Higher-order compact finite difference scheme with multigrid algorithm is applied in this paper for solving one-dimensional and two-dimensional inhomogeneous Helmholtz equations. In two-dimensional case, the suggested scheme has the stencil of twenty one points. An efficient solver multigrid method yields eighth-order accurate approximation on both fine and coarse grids. For the Neumann boundary condition, an eighth-order accurate representation is also developed. The accuracy and efficiency of eighth-order compact difference scheme are exhibited through graphical illustrations and computed results are drafted in tabular form to validate the numerical experiments.

Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations

The classical continuous finite element method with Lagrangian $Q^k$ basis reduces to a finite difference scheme when all the integrals are replaced by the $(k+1)\times (k+1)$ Gauss-Lobatto quadrature. We prove that this finite difference scheme is $(k+2)$-th order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values.

Entropy-stable schemes for relativistic hydrodynamics equations

In this article, we propose high-order finite difference schemes for the equations of relativistic hydrodynamics, which are entropy stable. The crucial components of these schemes are a computationally efficient entropy conservative flux and suitable high-order entropy dissipative operators. We first design a higher-order entropy conservative flux. For the construction of appropriate entropy dissipative operators, we derive entropy scaled right eigenvectors. This is then used with ENO-based sign-preserving reconstruction of scaled entropy variables, which results in higher-order entropy-stable schemes. Several numerical results are presented up to fourth order to demonstrate entropy stability and performance of these schemes.

A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system

This paper modifies a n-dimensional Hopf-Cole transformation to the n-dimensional Burgers’ system. We obtain the n-dimensional heat conduction equation through the modification of the Hopf-Cole transformation. Then the fourth-order precise integration method (PIM) in combination with a spatially global sixth-order compact finite difference (CFD) scheme is presented to solve the equation with high accuracy. Moreover, coupling with the Strang splitting method, the scheme is extended to multi-dimensional (two, three-dimensional) Burgers’ system. Numerical results show that the proposed method appreciably improves the computational accuracy compared with the existing numerical method. Moreover, the two-dimensional and three-dimensional examples demonstrate excellent adaptability, and the numerical simulation results also have very high accuracy in medium Reynolds numbers.

Using IIR filters to build high-order finite difference schemes for the unsteady Schrödinger equation

High-order finite difference schemes for the time-dependent Schrödinger equation are investigated. Digital signal processing methods allowed proving the conservativeness of high-order finite difference schemes for the unsteady Schrödinger equation. The eighth-order scheme coefficients were found with the help of the proved theoretical results. The conditions for equivalence between the eighth-order finite difference scheme and the scheme in the form of a cascade of allpass first-order filters were found. The numerical analysis of the proposed scheme was made. It was shown that the high-order finite difference schemes gave better results on solving the linear Schrödinger equations comparing to the well-known fourthorder scheme on the six-point stencil, however, the high-order schemes in couple with the second-order splitting algorithm to the nonlinear Schrödinger equation do not lead to a radical improvement in the quality of numerical results. Practical issues implementing the proposed numerical technique are considered. The obtained results can be used to construct efficient solvers for linear and nonlinear Schrödinger-type equations by applying the splitting schemes of adequate accuracy order.

A new high-order compact finite difference scheme based on precise integration method for the numerical simulation of parabolic equations

This paper presents two high-order exponential time differencing precise integration methods (PIMs) in combination with a spatially global sixth-order compact finite difference scheme (CFDS) for solving parabolic equations with high accuracy. One scheme is a modification of the compact finite difference scheme of precise integration method (CFDS-PIM) based on the fourth-order Taylor approximation and the other is a modification of the CFDS-PIM based on a (4,4)-Padé approximation. Moreover, on coupling with the Strang splitting method, these schemes are extended to multi-dimensional problems, which also have fast computational efficiency and high computational accuracy. Several numerical examples are carried out in order to demonstrate the performance and ability of the proposed schemes. Numerical results indicate that the proposed schemes improve remarkably the computational accuracy rather than the empirical finite difference scheme. Moreover, these examples show that the CFDS-PIM based on the fourth-order Taylor approximation yields more accurate results than the CFDS-PIM based on the (4,4)-Padé approximation.

Entropy Symmetrization and High-Order Accurate Entropy Stable Numerical Schemes for Relativistic MHD Equations

This paper presents entropy symmetrization and high-order accurate entropy stable schemes for the relativistic magnetohydrodynamic (RMHD) equations. It is shown that the conservative RMHD equations are not symmetrizable and do not admit a thermodynamic entropy pair. To address this issue, a symmetrizable RMHD system, equipped with a convex thermodynamic entropy pair, is proposed by adding a source term into the equations, providing an analogue to the non-relativistic Godunov--Powell system. Arbitrarily high-order accurate entropy stable finite difference schemes are developed on Cartesian meshes based on the symmetrizable RMHD system. The crucial ingredients of these schemes include (i) affordable explicit entropy conservative fluxes which are technically derived through carefully selected parameter variables, (ii) a special high-order discretization of the source term in the symmetrizable RMHD system, and (iii) suitable high-order dissipative operators based on essentially non-oscillatory reconstruction to ensure the entropy stability. Several numerical tests demonstrate the accuracy and robustness of the proposed entropy stable schemes.

Effects of compressibility and Atwood number on the single-mode Rayleigh-Taylor instability

In order to study the effect of compressibility on Rayleigh-Taylor (RT) instability, we numerically simulated the late-time evolution of two-dimensional single-mode RT instability for isothermal background stratification with different isothermal Mach numbers and Atwood numbers (At) using a high-order central compact finite difference scheme. It is found that the initial density stratification caused by compressibility plays a stabilizing role, while the expansion-compression effect of flow plays a destabilizing role. For the case of small Atwood number, the density difference between the two sides of the interface is small, and the density distribution of the upper and lower layers is nearly symmetrical. The initial density stratification plays a dominant role, and the expansion-compression effect has little influence. With the increase in the Atwood number, the stabilization effect of initial density stratification decreases, and the instability caused by the expansion-compression effect becomes more significant. The flow structures of bubbles and spikes are quite different at medium Atwood number. The effect of compressibility on the bubble velocity is strong at large At. The bubble height is approximately a quadratic function of time at potential flow growth stage. The average bubble acceleration is nearly proportional to the square of Mach number at At = 0.9.

High-Order Finite Difference Schemes Based on Symmetric Conservative Metric Method: Decomposition, Geometric Meaning and Connection with Finite Volume Schemes

High-Order Finite Difference Schemes Based on Symmetric Conservative Metric Method: Decomposition, Geometric Meaning and Connection with Finite Volume Schemes

High-order bound-preserving finite difference methods for miscible displacements in porous media

In this paper, we develop high-order bound-preserving (BP) finite difference (FD) methods for the coupled system of compressible miscible displacements. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component, cj, should be between 0 and 1. It is well known that cj does not satisfy a maximum-principle. Hence most of the existing BP techniques cannot be applied directly. The main idea in this paper is to construct the positivity-preserving techniques to all cj′s and enforce ∑jcj=1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation and choose suitable “consistent” numerical fluxes in the pressure and concentration equations. Recently, the high-order BP discontinuous Galerkin (DG) methods for miscible displacements were introduced in [4]. However, the BP technique for DG methods is not straightforward extendable to high-order FD schemes. There are two main difficulties. Firstly, it is not easy to determine the time step size in the BP technique. In finite difference schemes, we need to choose suitable time step size first and then apply the flux limiter to the numerical fluxes. Subsequently, we can compute the source term in the concentration equation, leading to a new time step constraint that may not be satisfied by the time step size applied in the flux limiter. Therefore, it would be very difficult to determine how large the time step is. Secondly, the general treatment for the diffusion term, e.g. centered difference, in miscible displacements may require a stencil whose size is larger than that for the convection term. It would be better to construct a new spatial discretization for the diffusion term such that a smaller stencil can be used. In this paper, we will solve both problems. We first construct a special discretization of the convection term, which yields the desired approximations of the source. Then we can find out the time step size that suitable for the BP technique and apply the flux limiters. Moreover, we will also construct a special algorithm for the diffusion term whose stencil is the same as that used for the convection term. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.

An efficient and high accuracy finite-difference scheme for the acoustic wave equation in 3D heterogeneous media

In this paper we developed a new explicit compact high-order finite difference scheme to solve the 3D acoustic wave equations with spatially variable acoustic velocity. The boundary conditions for the second derivatives of spatial variables have been derived by using the wave equation and the boundary conditions themselves. Theoretical analysis shows that the new scheme has an accuracy order of O(τ2) + O(h4), where τ is the time step and h is the grid size. Combined with Richardson extrapolation or Runge–Kutta method, the new method can be improved to 4th-order accuracy in time, which has been implemented and verified in this paper. Four numerical experiments are conducted to validate the efficiency and accuracy of the new scheme. The stability of the new scheme has been proved by an energy method, which shows that the new scheme is conditionally stable with a Currant–Friedrichs–Lewy (CFL) number which is slightly lower than that of the Padé approximation based method. However, the new scheme is much simpler to implement.