Compact Finite Difference Scheme Research Articles

High-order compact finite difference schemes for the time-fractional Black-Scholes model governing European options

Based on the price fluctuations of the underlying fractal transmission system, the α-order time-fractional Black-Scholes model is obtained. In this paper, we introduce two compact finite difference schemes for the solution of the time-fractional Black-Scholes equation governing European option pricing. In proposed schemes, in order to gain sixth-order and eighth-order accuracy in space, we first use exponential transformation to eliminate the convection term of the Black-Scholes equation. Then, the time-fractional derivative discretizes by a 3−αth order numerical formula (called the L1–2 formula here) which is constructed by a linear interpolating polynomial on the first subinterval and the quadratic interpolating polynomials on the other subintervals. We investigate stability and convergence of proposed schemes by Fourier method. Finally, some numerical examples perform to demonstrate the theoretical order of accuracy and illustrate the effectiveness of proposed methods. We also discuss the influence of different parameters on the option price in the time-fractional Black-Scholes model.

Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme

In this paper, a new numerical scheme based on weighted and shifted Grünwald formula and compact difference operate is proposed. The proposed numerical scheme is used to solve time fractional partial integro-differential equation with a weakly singular kernel. Meanwhile the time fractional derivative is denoted by the Riemann-Liouville sense. Subsequently, we prove the stability and convergence of the mentioned numerical scheme and show that the numerical solution converges to the analytical solution with order O(τ2 + h4), where τ and h are time step size and space step size, respectively. The advantage is that the accuracy of the suggested schemes is not dependent on the fractional α. Furthermore, the numerical example shows that the method proposed in this paper is effective, and the calculation results are consistent with the theoretical analysis.

A reconstructed discontinuous Galerkin method based on variational formulation for compressible flows

A new reconstructed discontinuous Galerkin (rDG) method based on variational formulation is developed for compressible flows. In the presented method, a higher-order piece-wise polynomial is reconstructed based on the underlying discontinuous Galerkin (DG) solution. This reconstruction is done by using a newly developed variational formulation. The variational reconstruction (VR) can be seen as an extension of the compact finite difference (FD) schemes to unstructured grids. The reconstructed variables are obtained by solving an extreme-value problem, which minimizes the jumps of the reconstructed piece-wise polynomials across the cell interfaces, and therefore maximizes the smoothness of the reconstructed solution. Intrinsically, the stencils of the presented reconstruction are the entire mesh, so this method is robust even on tetrahedral grids. A variety of benchmark test cases are presented to assess the accuracy, efficiency and robustness of this rDG method. The numerical experiments demonstrate that the developed rDG method based on variational formulation can maintain the linear stability, obtain the designed high-order accuracy, and outperform the rDG counterpart based on the least-squares reconstruction for both inviscid and viscous compressible flows.

High-order compact finite difference scheme with two conserving invariants for the coupled nonlinear Schrödinger–KdV equations

In this paper, a new high-order accurate compact finite difference scheme for the coupled nonlinear Schrödinger–KdV (CNLS–KdV) equations is proposed. Conservation of the discrete number of plasmon and the discrete number of particle are given in detail. Convergence with second-order in time and fourth-order in space of the present scheme are proved by using “cut-off” function technique and discrete energy method. Numerical experiments are given to support the theoretical analysis.

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

Uniform error bound of a conservative fourth-order compact finite difference scheme for the Zakharov system in the subsonic regime

A high order compact finite difference scheme for elliptic interface problems with discontinuous and high-contrast coefficients

The elliptic interface problems with discontinuous and high-contrast coefficients appear in many applications and often lead to huge condition numbers of the corresponding linear systems. Thus, it is highly desired to construct high order schemes to solve the elliptic interface problems with discontinuous and high-contrast coefficients. Let Γ be a smooth curve inside a rectangular region Ω. In this paper, we consider the elliptic interface problem −∇·(a∇u)=f in Ω∖Γ with Dirichlet boundary conditions, where the coefficient a and the source term f are smooth in Ω∖Γ and the two jump condition functions [u] and [a∇u·n→] across Γ are smooth along the interface Γ. To solve such elliptic interface problems, we propose a high order compact 9-point finite difference scheme and a high order local calculation for numerically computing the solution u and its gradient ∇u respectively on uniform Cartesian grids without changing coordinates into local coordinates. We numerically verify the sign conditions of our proposed compact finite difference scheme and prove the convergence rate by the discrete maximum principle. Our numerical experiments confirm the fourth order accuracy for computing the solution u in both l2 and l∞ norms of the proposed compact finite difference scheme on uniform meshes for the elliptic interface problems with discontinuous and high-contrast coefficients.

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

A computational source modeling of brain activity: An inverse problem

Source determination in an inverse problem from the over-specified data plays a crucial role in cognitive modeling. In this paper, an accurate and fast method is proposed for solving the one-dimensional inverse problem concerning diffusion equation with source control parameter. The proposed method is based on applying a compact finite difference scheme for spatial components and solving the system that arose from this scheme by multigrid.

Heat Transfer Past a Rotationally Oscillating Circular Cylinder in Linear Shear Flow

Abstract This study investigates the unsteady, two-dimensional flow and heat transfer past a rotationally oscillating circular cylinder in linear shear flow. A higher order compact (HOC) finite difference scheme is used to solve the governing Navier–Stokes equations coupled with the energy equation on a nonuniform grid in polar coordinates. The hydrodynamic and thermal features of the flow are mainly influenced by the shear rate (K), Reynolds number (Re), Prandtl number (Pr), and the cylinder oscillation parameters, i.e., oscillation amplitude (αm), the frequency ratio (fr). The simulations are performed for Re=100,Pr=0.5−1.0, 0.0≤K≤0.15, and 0.5≤αm≤2.0. The numerical scheme is validated with the existing literature studies. Partial and full vortex suppression is observed for certain values of shear parameter K. The connection between heat transfer and vortex shedding phenomenon is examined where a pronounced increase in the heat transfer is observed for certain values of oscillation parameter, relative to the nonshear flow case.

A new three-level fourth-order compact finite difference scheme for the extended Fisher-Kolmogorov equation

In this paper, a new fourth-order compact finite difference scheme is proposed to solve the extended Fisher-Kolmogorov (EFK) equation. The scheme is three-level and implicit nonlinear. A new time discretization technique is adopted in the new scheme. The priori estimate and convergence of the numerical solution are obtained by using the discrete energy method, and the optimal error estimation in the maximum norm shows that the proposed numerical scheme has second-order accuracy in time and fourth-order accuracy in space. Finally, the theoretical analysis is supported by the numerical results of several examples.

A fourth-order compact finite difference scheme for the quantum Zakharov system that perfectly inherits both mass and energy conservation

In this paper, we propose and analyze a new fourth-order compact finite difference scheme for solving the quantum Zakharov system (QZS). The new scheme perfectly inherits the mass and energy conservation possessed by the QZS, while the energy preserved by the existing schemes expressed by two-level's solution at each time step. By using the energy method to analyze an equivalent form of the difference scheme, without any requirement on the grid ratio, we establish rigorously the optimal error estimate of the proposed scheme in the energy norm. The accuracy is shown to be fourth order in space and second order in time. Numerical examples are presented to verify the theoretical results and show the effectiveness of the proposed scheme.

A Fourth-Order Compact Finite Difference Scheme for Solving the Time Fractional Carbon Nanotubes Model.

In this work, we deal with unsteady magnetohydrodynamic allowed convection inflow of blood with a carbon nanotubes model; the single and multiwalled carbon nanotubes of human blood are used as a based fluid. Two numerical methods used to study this model are the weighted average finite difference method and the nonstandard compact finite difference method. The proportional Caputo hybrid operator has been used to fractionalize the proposed model. Stability analysis has been construed by a kind of John von Neumann stability analysis. Numerical results are presented in diverse graphs, which manifest that the method is successful in solving the proposed model.

Compact Finite Difference Scheme Combined with Richardson Extrapolation for Fisher’s Equation

In this study, the fourth-order compact finite difference scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation is presented. First, the derivative involving the space variable is discretized by the fourth-order compact finite difference method. Then, the nonlinear term is linearized by the lagging method, and the derivative involving the temporal variable is discretized by the Crank–Nicolson scheme. The method is found to be unconditionally stable and fourth-order accurate in the direction of the space variable and second-order accurate in the direction of the temporal variable. When combined with the Richardson extrapolation, the order of the method is improved from fourth to sixth-order accurate in the direction of the space variable. The numerical results displayed in figures and tables show that the proposed method is efficient, accurate, and a good candidate for solving the 1D Fisher’s equation.

High order approximation on non-uniform meshes for generalized time-fractional telegraph equation

This paper presents a high order approximation scheme to solve the generalized fractional telegraph equation (GFTE) involving the generalized fractional derivative (GFD). The GFD is characterized by a scale function σ(t) and a weight function ω(t). Thus, we study the solution behavior of the GFTE for different σ(t) and ω(t). The scale function either stretches or contracts the solution while the weight function dramatically shifts the numerical solution of the GFTE. The time fractional GFTE is approximated using quadratic scheme in the temporal direction and the compact finite difference scheme in the spatial direction. To improve the numerical scheme’s accuracy, we use the non-uniform mesh. The convergence order of the whole discretized scheme is, O(τ2α−3,h4), where τ and h are the temporal and spatial step sizes respectively. The outcomes of the work are as follows: •The error estimate for approximation of the GFD on non-uniform meshes is established.•The numerical scheme’s stability and convergence are examined.•Numerical results for four examples are compared with those obtained using other method. The study shows that the developed scheme achieves higher accuracy than the scheme discussed in literature.

Hybrid Finite Difference Schemes for Elliptic Interface Problems with Discontinuous and High-Contrast Variable Coefficients

For elliptic interface problems with discontinuous coefficients, the maximum accuracy order for compact 9-point finite difference scheme in irregular points is three [7]. The discontinuous coefficients usually have abrupt jumps across the interface curve in the porous medium of realistic problems, causing the pollution effect of numerical methods. So, to obtain a reasonable numerical solution of the above problem, the higher order scheme and its effective implementation are necessary. In this paper, we propose an efficient and flexible way to achieve the implementation of a hybrid (9-point scheme with sixth order accuracy for interior regular points and 13-point scheme with fifth order accuracy for interior irregular points) finite difference scheme in uniform meshes for the elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle $\Omega$. We also derive the $6$-point and $4$-point finite difference schemes in uniform meshes with sixth order accuracy for the side points and corner points of various mixed boundary conditions (Dirichlet, Neumann and Robin) of elliptic equations in a rectangle. Our numerical experiments confirm the flexibility and the sixth order accuracy in $l_2$ and $l_{\infty}$ norms of the proposed hybrid scheme.

Two different temporal domain integration schemes combined with compact finite difference method to solve modified Burgers’ equation

The modified Burgers’ equation has application in various fields such as explosions and sonic boom theory, propagation of torsional waves in thin circular viscoelastic rod, acoustic waves produced by laser radiations, etc. A sixth-order compact finite difference scheme has been used for the space integration. Two different approaches have been followed for time integration. One is the Crank-Nicolson scheme in which the nonlinear term is handled with a quasilinearization process. Another is the strong stability preserving Runge-Kutta method of four stage, third-order convergence (SSP-RK43), in which no linearization is required. The stability analysis for both the schemes has been presented and is shown to be stable. Few test problems have been solved, demonstrating the performance of the proposed techniques. The L2 and L∞ error norms are calculated and are compared with the previous work. The aim of the paper is to solve such an important equation with better accuracy and less computational efforts.

A class of HOC finite difference method for elliptic interface problems with imperfect contact

<abstract><p>The elliptic interface problems with imperfect contact have found applications in numerical solutions of the Stefan problem of the solidification process and crystal growth, composite materials, multi-phase flows, etc. In this paper a 1D elliptic interface problem with imperfect contact is considered. A class of high-order compact finite difference schemes are constructed on body-fitted and non-body-fitted mesh, respectively. For each case, the second-, third- and fourth-order approximations of implicit jump conditions are provided by using the jump conditions and its high-order derivatives. Numerical examples are provided to verify the performance of the schemes. The numerical results demonstrate that the schemes have theoretical accuracy for elliptic interface problems with imperfect contact.</p></abstract>

Heat transfer past rotationally oscillating circular cylinder heated with time‐periodic pulsating temperature in a uniform flow

AbstractThe transient forced convection heat transfer from an impulsively started rotationally oscillating circular cylinder, heated with time‐periodic pulsating temperature in a two‐dimensional uniform, laminar and viscous flow of Newtonian, incompressible, constant property fluid is investigated in this paper. The two‐dimensional Navier‐Stokes and energy equations are solved using a higher‐order compact (HOC) finite difference scheme on non‐uniform polar grids. Computational code for solving the problem is written in C programming language and is validated against existing research works. The flow and heat transfer characteristics are mainly influenced by the amplitude of time‐periodic pulsating temperature (), period of time‐periodic pulsating temperature (), Prandtl number (), Reynolds number (), and cylinder oscillation parameters i.e. maximum rotational rate (), frequency ratio (). The numerical computations are done for the range of parameter values , , , , and . The backward heat transfer is observed for certain values of the amplitude of time‐periodic pulsating temperature and the amount of backward heat transfer enhances with increasing amplitude of temperature. High period value () of pulsating temperature is found to reduce the amount of heat convection downstream. The influence of pulsating temperature is discussed for an increase or decrease in heat transfer rate relative to the constant cylinder‐temperature case.

A diffuse-interface compact difference method for compressible multimaterial elastic–plastic flows

In this paper, a novel high-order numerical method is proposed for the simulation of compressible multimaterial solid–fluid problems involving elastic–plastic solid behaviors. The solid, liquid, and gas behaviors are described by an Eulerian interface-capturing model derived from hyperelasticity theory. The deformations in each component of the solids are tracked using a single deviatoric strain tensor. The numerical algorithm employs a tenth-order compact finite difference scheme and a fourth-order Runge–Kutta time-stepping scheme. To prevent numerical oscillations near the material interface, a mechanical equilibrium assumption is introduced for the elastic–plastic problems, and a localized artificial diffusivity method satisfying this assumption is designed and applied in the numerical algorithm. Numerical tests in one and two dimensions are shown to demonstrate the accuracy and feasibility of the proposed approach. In particular, the accuracy and numerical resolution of the method are verified using problems with analytical solutions. Numerical tests of interface advection problems demonstrate that our method preserves the velocity, pressure, and elastic stress equilibria at the material interface and prevents overshoots in the rest of the domain. The two-dimensional Richtmyer–Meshkov instabilities between two elastic or two elastic–plastic solids are simulated to verify the suitability of the numerical method for solving problems involving large deformations. Impact of a 2D Taylor bar to a rigid wall is also simulated to test the robustness of the numerical method.