Sixth-order Finite Difference Research Articles

A sixth-order finite difference method for the two-dimensional nonlinear advection diffusion reaction equation

A novel higher-order finite difference method is developed to solve the two-dimensional nonlinear advection diffusion reaction equation. Firstly, a new five-point sixth-order finite difference scheme is proposed to discretize spatial derivatives and the sixth-order backward difference method is applied to teat temporal derivative. As a result, a new fully implicit scheme with sixth-order accuracy in both time and space is proposed. Subsequently, for the calculation of the starting step of the sixth-order backward difference method, the Crank-Nicolson method combined with the Richardson extrapolation technique is employed to obtain the equivalent temporal accuracy as the main scheme. Finally, the accuracy and reliability of the presented method are validated through various numerical examples.

Exploring the Formation of Resistive Pseudodisks with the GPU Code Astaroth

Pseudodisks are dense structures formed perpendicular to the direction of the magnetic field during the gravitational collapse of a molecular cloud core. Numerical simulations of the formation of pseudodisks are usually computationally expensive with conventional CPU codes. To demonstrate the proof of concept of a fast computing method for this numerically costly problem, we explore the GPU-powered MHD code Astaroth, a sixth-order finite difference method with low adjustable finite resistivity implemented with sink particles. The formation of pseudodisks is physically and numerically robust and can be achieved with a simple and clean setup for this newly adopted numerical approach for science verification. The method’s potential is illustrated by evidencing the dependence on the initial magnetic field strength of specific physical features accompanying the formation of pseudodisks, e.g., the occurrence of infall shocks and the variable behavior of the mass and magnetic flux accreted on the central object. As a performance test, we measure both weak and strong scaling of our implementation to find the most efficient way to use the code on a multi-GPU system. Once suitable physics and problem-specific implementations are realized, the GPU-accelerated code is an efficient option for 3D magnetized collapse problems.

Sixth-order hybrid finite difference methods for elliptic interface problems with mixed boundary conditions

In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem −∇⋅(a∇u)=f in Ω﹨Γ, where Γ is a smooth interface inside Ω. The variable scalar coefficient a>0 and source f are possibly discontinuous across Γ. The hybrid FDMs utilize a 9-point compact stencil at any interior regular points of the grid and a 13-point stencil at irregular points near Γ. For interior regular points away from Γ, we obtain a sixth-order 9-point compact FDM satisfying the sign and sum conditions for ensuring the M-matrix property. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs satisfying the sign and sum conditions for the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. Thus, for the elliptic problem without interface (i.e., Γ is empty), our compact FDM has the M-matrix property for any mesh size h>0 and consequently, satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. For irregular points near Γ, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient a, the source term f, the interface curve Γ, the two jump functions along Γ, and the functions on ∂Ω. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing Γ and are also applicable if the jump conditions on Γ only depend on the geometry (e.g., curvature) of the curve Γ. Our numerical experiments confirm the sixth-order convergence in the l∞ norm of the proposed hybrid FDMs for the elliptic interface problem.

Sixth-order finite difference schemes for nonlinear wave equations with variable coefficients in three dimensions

First, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.

A sixth-order finite difference HWENO scheme for nonlinear degenerate parabolic equation

A sixth-order finite difference HWENO scheme for nonlinear degenerate parabolic equation

A sixth-order central WENO scheme for nonlinear degenerate parabolic equations

In this paper, we develop a new sixth-order finite difference central weighted essentially non-oscillatory (WENO) scheme with Z-type nonlinear weights for nonlinear degenerate parabolic equations. The centered polynomial is introduced for the WENO reconstruction to avoid the negative linear weights. We choose the Z-type nonlinear weights based on the $$L^2$$ -norm smoothness indicators, yielding the new WENO scheme with less computational cost. It is also confirmed that the proposed central WENO scheme with the devised nonlinear weights achieves sixth order accuracy in smooth regions. One- and two-dimensional numerical examples are presented to demonstrate the improved performance of the proposed central WENO scheme.

A sixth-order finite difference WENO scheme for non-linear degenerate parabolic equations: an alternative technique

PurposeThis paper aims to construct a sixth-order weighted essentially nonoscillatory scheme for simulating the nonlinear degenerate parabolic equations in a finite difference framework.Design/methodology/approachTo design this scheme, we approximate the second derivative in these equations in a different way, which of course is still in a conservative form. In this way, unlike the common practice of reconstruction, the approximation of the derivatives of odd order is needed to develop the numerical flux.FindingsThe results obtained by the new scheme produce less error compared to the results of other schemes in the literature that are recently developed for the nonlinear degenerate parabolic equations while requiring less computational times.Originality/valueThis research develops a new weighted essentially nonoscillatory scheme for solving the nonlinear degenerate parabolic equations in multidimensional space. Besides, any selection of the constants (sum equals one is the only requirement for them), named the linear weights, will obtain the desired accuracy.

A sixth-order finite difference method for solving the generalized Burgers-Fisher and generalized Burgers-Huxley equations

A sixth-order finite difference method for solving the generalized Burgers-Fisher and generalized Burgers-Huxley equations

Efficient sixth-order finite difference method for the two-dimensional nonlinear wave equation with variable coefficient

Efficient sixth-order finite difference method for the two-dimensional nonlinear wave equation with variable coefficient

A novel combination of mathematical modelling approaches for simulating nearly hypersonic, viscous, reacting magnetogasdynamic flows

This work focuses on the mathematical combination of magnetogasdynamics (MGD) and the Dunn and Kang chemical kinetic model with a multi-component Harten–Lax–van Leer Contact (HLLC) local Riemann solver using high-order interpolation schemes. Due to the nature of nearly hypersonic, unsteady, viscous, reacting MGD flows, the gas dissociates which is taken into account with 11 species and 26 chemical reactions along with the temperature dependent transport properties through the Wilke model. In the present work, the electromagnetic effects are considered by solving a Poisson equation simultaneously with the chemical reaction equations. The non-linear convective terms are treated numerically with third-, fifth- and seventh-order Weighted Essentially Non-Oscillatory (WENO) interpolation schemes along with second-, fourth- and sixth-order central finite difference schemes for the viscous terms. The combination of these numerical and physical models are analysed for stability and validated for benchmark problems such as (a) Hartmann flow as MGD validation test case and (b) the shock-tube problem for testing the chemical reaction model in which case the reaction rate controlling temperature suggested by Park has also been taken into account. Furthermore, a complex physical problem of an external aerodynamic flow over the cube for the Mach number varying from 1.25 to 3 have also been investigated, because a limited number of studies is available for understanding the flow behaviour of sharp cornered blunt bodies such as cubes. The knowledge of shock structures of these objects has become imperative to predict the atmospheric re-entry trajectories of de-orbiting CubeSats, space debris or fragments of launch vehicles. The main physical finding is that the shock structures are observed to be much more resolved compared to previous works, especially in the wake region. The knowledge gap is bridged by detailing the shock structures and showing the dependence of the shock standoff distance on total enthalpy of the flow.

Valuation of the American put option as a free boundary problem through a high-order difference scheme

Abstract Valuation of the American options encountered commonly in finance is quite difficult due to the possibility of early exercise alternatives. Since an exact solution for the American options does not exist, effective numerical methods are needed to understand the behavior of option pricing models. Therefore, in this paper, a new approach based on a high-order difference scheme is proposed to discuss the valuation of an American put option as a free boundary problem. Using a front-fixing approach that transforms the unknown free boundary (optimal stopping) into a fixed one, a sixth-order finite difference scheme (FD6) in space and a third-order strong-stability preserving Runge–Kutta (SSPRK3) in time are applied to the model converted to a nonlinear partial differential equation. The computed results revealed that the combined method is seen to attempt to pull up the capacity of the algorithm to achieve higher accuracy. It is seen that the quantitative and qualitative results produced by the method proposed with minimal computational effort are sufficiently accurate and meaningful. Therefore, this article provides some new insights about the physical characteristics of financial problems and such realistic phenomena.

A New Sixth-Order Finite Difference WENO Scheme for Fractional Differential Equations

In this paper, we propose a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. After splitting the Caputo fractional derivative of order $$\alpha $$ $$(1<\alpha \le 2)$$ into a weakly singular integral and a classical second derivative, the classical Gauss–Jacobi quadrature is used to solve the weakly singular integral and a new spatial WENO-type reconstruction methodology is proposed to approximate the second derivative. There are two advantages of the new WENO scheme: the first is that the linear weights can be any positive numbers on condition that their summation equals one, and the second is its simplicity in the engineering applications. The new WENO reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on three unequal-sized spatial stencils in a traditional WENO fashion. This new sixth-order WENO scheme uses smaller number of cell average information than that of the same order classical WENO schemes and could eliminate non-physical oscillations near strong discontinuities when solving the fractional differential equations. Some benchmark examples are given to demonstrate the efficiency, robustness, and good performance of this new finite difference WENO scheme.

Sixth Order Finite Difference Method for Three Coupled Nonlinear Schr&amp;#246;dinger Equations

In this work, we will derive a numerical method of sixth order in space and second order in time for solving 3-coupled nonlinear Schrödinger equations. The numerical method is unconditionally stable. We use the exact single soliton solution and the conserved quantities to check the accuracy and the efficiency of the proposed schemes. Also, we study the interaction dynamics of two solitons. It is found that both elastic and inelastic collisions can take place under suitable parametric conditions.

L1-type smoothness indicators based WENO scheme for nonlinear degenerate parabolic equations

In this article an efficient sixth-order finite difference weighted essentially non-oscillatory scheme is developed to solve nonlinear degenerate parabolic equations. A new type of nonlinear weights are constructed with an introduction of a global smoothness indicator by a linear combination of local derivatives information involved in the smaller stencils that are developed with the help of generalized undivided differences in L1-norm. The advantage with these local smoothness indicators is that, each derivative involved in these are of higher order accurate as compared to the smoothness indicators developed by Liu et al. [SIAM, J. Sci. Comput, 2011]. Based on method of lines, we use strong stability preserving third-order Runge-Kutta (SSP-RK) scheme to evaluate time derivative. Various numerical tests are conducted in one and two dimensions to demonstrate the performance enhancement, resolution power and numerical accuracy of the scheme. The proposed scheme provides a better agreement in achieving the numerical solution as compared to the schemes developed by Liu et al. [SIAM, J. Sci. Comput., 2011] and Hajipour and Malek [Appl. Math. Model., 2012].

Numerical simulation of flow-induced vibration of two cylinders elastically mounted in tandem by immersed moving boundary method

A numerical study is performed on flow-induced vibrations of two cylinders of diameter D, in tandem configuration relative to the free-stream uniform flow U at low Reynolds numbers Re=UD/ν. In order to solve numerically the incompressible momentum and continuity equations, the in-house code Incompact3D is used. Compact sixth-order finite differences for spatial differentiation and second-order Adams–Bashforth scheme for time advancement are employed. The cylinders movement is modeled as a mass-damper-spring system which is solved by a fourth-order Runge–Kutta scheme. To represent the multiple moving cylinders in an immersed boundary method framework, a modification in the Poisson equation is proposed. The modified algorithm was evaluated and applied for scenarios with zero, one and two translational degrees of freedom to oscillate. The validation is carried out against several numerical and experimental previous works. The results are analyzed in terms of the forces on cylinders, Strouhal number of the wake, streamwise velocity profiles and cylinders oscillation amplitudes and frequencies. The algorithm could satisfactorily represent the resonance and wake-flutter phenomena. In the most cases, the rear cylinder has greater oscillations than the front cylinder. However, a state was identified where the cross-stream oscillation amplitudes of the front cylinder are lower than those of the rear cylinder. In this state, the lift force acts as negative spring on the front cylinder (amplifying its oscillations) and as a damper on the rear cylinder (controlling its oscillations). The scenarios with two degrees of freedom (2dof) have higher oscillation amplitudes than the equivalent scenarios with one degree of freedom. Moreover, the streamwise oscillation amplitudes are not negligible in relation to the cross-stream oscillation amplitudes. On the other hand, for 2dof, when the Reynolds number increases the clashing risk also increases, since the cylinders proximity is narrowed.

Sixth-order finite difference scheme for the Helmholtz equation with inhomogeneous Robin boundary condition

In this paper, a class of sixth-order finite difference schemes for the Helmholtz equation with inhomogeneous Robin boundary condition is derived. This scheme is based on the sixth-order approximation for the Robin boundary condition by using the Helmholtz equation and the Taylor expansion, by which the ghost points in the scheme on the domain can be eliminated successfully. Some numerical examples are shown to verify its correctness and robustness with respect to the wave number.

The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence

ABSTRACTHydromagnetic turbulence produced during phase transitions in the early universe can be a powerful source of stochastic gravitational waves (GWs). GWs can be modelled by the linearised spatial part of the Einstein equations sourced by the Reynolds and Maxwell stresses. We have implemented two different GW solvers into the Pencil Code – a code which uses a third order timestep and sixth order finite differences. Using direct numerical integration of the GW equations, we study the appearance of a numerical degradation of the GW amplitude at the highest wavenumbers, which depends on the length of the timestep – even when the Courant–Friedrichs–Lewy condition is ten times below the stability limit. This degradation leads to a numerical error, which is found to scale with the third power of the timestep. A similar degradation is not seen in the magnetic and velocity fields. To mitigate numerical degradation effects, we alternatively use the exact solution of the GW equations under the assumption that the source is constant between subsequent timesteps. This allows us to use a much longer timestep, which cuts the computational cost by a factor of about ten.

A New Approach for the Black–Scholes Model with Linear and Nonlinear Volatilities

Since financial engineering problems are of great importance in the academic community, effective methods are still needed to analyze these models. Therefore, this article focuses mainly on capturing the discrete behavior of linear and nonlinear Black–Scholes European option pricing models. To achieve this, this article presents a combined method; a sixth order finite difference (FD6) scheme in space and a third–order strong stability preserving Runge–Kutta (SSPRK3) over time. The computed results are compared with available literature and the exact solution. The computed results revealed that the current method seems to be quite strong both quantitatively and qualitatively with minimal computational effort. Therefore, this method appears to be a very reliable alternative and flexible to implement in solving the problem while preserving the physical properties of such realistic processes.

A fast preconditioned iterative method for the electromagnetic scattering by multiple cavities with high wave numbers

We study the problem of electromagnetic scattering by multiple open cavities embedded in an infinite ground plane with high wave numbers. The problem can be described by a series of Helmholtz equations with coupled boundary conditions. We develop a sixth-order finite difference scheme to discretize the coupled Helmholtz equations. By Gaussian elimination in the vertical direction and Fourier transform in the horizontal direction, we can reduce the multiple cavity scattering problem to an aperture linear system. However, in the situation of high wave numbers, the condition number of the coefficient matrix of the reduced linear system is especially large and the system tends to be ill-conditioned. The convergence histories of most iterative methods become oscillating which consume considerable computations and memory spaces. In order to overcome the difficulty caused by high wave numbers, we develop an efficient preconditioned iterative method based on the Krylov subspace, which greatly improves the eigenvalue distributions and reduces the number of iterations. Numerical experiments show the validity and efficiency of the proposed sixth-order fast preconditioned algorithm for solving the scattering by multiple cavities with high wave numbers.

Cross-rhombus stencil-based finite-difference methods for 2D acoustic modeling and reverse-time migration on rectangular grids

The recently developed cross-rhombus stencil-based time–space domain finite-difference (FD) method for modeling two-dimensional acoustic equations, controls the temporal and spatial dispersions synchronously and outperforms cross-stencil-based FD (CS-FD). However, it is only applicable to modeling on equally spaced grids and extensive application is hindered. In this work, we extend it further and develop novel cross-rhombus stencil-based FD (CRS-FD) for modeling on arbitrarily rectangular grids. Two kinds of rhombus stencils involving grid points both on and off the axis are developed first to achieve fourth order and sixth order FD accuracies, respectively, for solving the second order temporal derivative on rectangular grids. The plane wave theory is then used to derive the time–space-domain dispersion relations, when the temporal and spatial derivatives are solved by the new rhombus-stencil-based temporal high order FD and the cross-stencil-based spatial high order FD, respectively. The extrapolation stencil is a mixture of rhombus and cross stencils, and the involved FD coefficients are determined by applying the Tayler expansion on the time–space domain dispersion relation. Our new CRS-FD is high-order accurate in both time and space, and applicable to modeling on arbitrarily rectangular grids. Dispersion analysis, stability analysis and modeling examples on rectangular grids show that the CRS-FD is more accurate and stable than the CS-FD. Meanwhile, we develop the variable-length schemes for CRS-FD to further increase efficiency and apply them to extrapolate wavefields in reverse time migration. The results validate that the CRS-FD is more efficient than the CS-FD because much larger time steps can be used while reaching a similar accuracy. The variable-length scheme further reduces the computational time in comparison with the fixed-length scheme.