Accurate Finite Difference Scheme Research Articles

High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes

This paper develops high-order accurate well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers M⩾2) shallow water equations (SWEs) with non-flat bottom topography on both fixed and adaptive moving meshes, extending our previous work on the single-layer shallow water magnetohydrodynamics [25] and single-layer SWEs on adaptive moving meshes [58]. To obtain an energy inequality, the convexity of an energy function for an arbitrary M is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into M identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit strong stability preserving Runge-Kutta methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [58]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.

Second-order numerical method for a neutral Volterra integro-differential equation

This paper is dedicated to obtaining an approximate solution for a neutral second-order Volterra integro-differential equation. Our method is the second-order accurate finite difference scheme on a uniform mesh. The error analysis is carried out and numerical results are given to support the proposed approach.

New higher-order accurate super-compact scheme for three-dimensional natural convection and entropy generation

Exploring natural convection in three dimensions through numerical analysis has become essential for gaining a deeper understanding of this process compared to studying it in just two dimensions. This paper presents an enhanced analysis of three-dimensional (3D) natural convection phenomena within an air-filled cubical cavity, with a primary focus on heat transfer characteristics. The distinguishing feature of this research lies in the pioneering application of the newly developed higher-order accurate super-compact finite difference scheme to study 3D natural convection and entropy generation. This numerical method is renowned for its fourth-order spatial accuracy and second-order temporal accuracy. The super-compact scheme employs 19 grid points at the current time level (nth time level) while utilizing just seven grid points from the subsequent time level [(n+1)th time level. The flow is considered to be 3D, time-varying, laminar, and incompressible. First, the newly developed numerical code has undergone validation through quantitative and qualitative comparisons with existing results. Then, we analyze the flow phenomena and heat transfer dynamics within the cavity for various Rayleigh numbers (102≤Ra≤105) with a fixed Prandtl number (Pr = 0.71) by using this scheme. Various parameters such as local Nusselt numbers, averaged Nusselt numbers, local entropy, Bejan number, streamlines, and isotherm dispersion are studied in this work. It is found that as the Rayleigh number (Ra) increases, the Bejan number (Be) decreases, while the total entropy increases. When the Ra is less than or equal to 104, the Be remains above 0.5, indicating that irreversibility primarily arises from heat transfer. However, as we transition to Ra=105, the Be falls below 0.5, signaling that irreversibility is now predominantly driven by viscous effects. To the best of our knowledge, this research marks a pioneering effort by introducing the application of the higher order super-compact (HOSC) scheme for the examination of heat transfer within a 3D cubic cavity, thus endowing our work with a truly novel and pioneering character. It is worth mentioning that, before this study, there has been no precedent in the exploration of heat transfer using the super-compact scheme, thus endowing our work with a truly novel and pioneering character.

High-Order, Accurate Finite Difference Schemes for Fourth-Order Differential Equations

High-Order, Accurate Finite Difference Schemes for Fourth-Order Differential Equations

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.

On the Accuracy of Finite-Difference Schemes in Calculations of Centered Rarefaction Waves

On the Accuracy of Finite-Difference Schemes in Calculations of Centered Rarefaction Waves

On accuracy of finite-difference schemes in calculations of centered rarefaction waves

Проведен сравнительный анализ точности разностных схем UpWind (первого порядка), TVD (второго порядка) и WENO5 (третьего порядка по времени) при расчете для нелинейного уравнения переноса задачи Коши с кусочно-линейными разрывными периодическими начальными данными. Показано, что в случае устойчивых начальных разрывов, из которых формируется последовательность ударных волн, порядок сходимости всех трех схем между ударными волнами совпадает с их формальной точностью. В случае неустойчивых начальных разрывов, при распаде которых формируется последовательность центрированных волн разрежения, все три схемы имеют первый порядок сходимости внутри этих волн. Получена явная формула для дисбалансов разностных решений в центрированной волне разрежения. В случае схем повышенной точности эта формула хорошо согласуется с численными расчетами, не зависит от типа схемы и определяется погрешностью аппроксимации начальных данных в окрестности неустойчивого сильного разрыва.

On the reduction in accuracy of finite difference schemes on manifolds without boundary

Abstract We investigate error bounds for numerical solutions of divergence structure linear elliptic partial differential equations (PDEs) on compact manifolds without boundary. Our focus is on a class of monotone finite difference approximations, which provide a strong form of stability that guarantees the existence of a bounded solution. In many settings including the Dirichlet problem, it is easy to show that the resulting solution error is proportional to the formal consistency error of the scheme. We make the surprising observation that this need not be true for PDEs posed on compact manifolds without boundary. We propose a particular class of approximation schemes built around an underlying monotone scheme with consistency error $O(h^{\alpha })$. By carefully constructing barrier functions, we prove that the solution error is bounded by $O(h^{\alpha /(d+1)})$ in dimension $d$. We also provide a specific example where this predicted convergence rate is observed numerically. Using these error bounds, we further design a family of provably convergent approximations to the solution gradient.

A high order accurate numerical algorithm for the space-fractional Swift-Hohenberg equation

This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). Based on the operator splitting method, the original problem is split into linear and nonlinear subproblems, respectively. The Fourier spectral method is adopted for the linear part, and a first-order accurate finite difference scheme in time together with the Fourier spectral method in space is used for the nonlinear part. The stability and convergence of the obtained numerical scheme are analyzed theoretically. Moreover, the spectral deferred correction (SDC) method is then employed to improve the temporal accuracy. Various two and three dimensional numerical experiments are performed to validate the theoretical results and efficiency of the proposed method.

An Accurate Compact Finite Difference Scheme for Solving the American Option with M-Regime Switching Model

An Accurate Compact Finite Difference Scheme for Solving the American Option with M-Regime Switching Model

An efficient fourth-order accurate conservative scheme for Riesz space fractional Schrödinger equation with wave operator

In this study, we investigate a high-order accurate conservative finite difference scheme by utilizing a fourth-order fractional central finite difference method for the two-dimensional Riesz space-fractional nonlinear Schrödinger wave equation. The conservation laws of the discrete difference scheme are shown. Meanwhile, the exactness, uniqueness, and prior estimate of the numerical solution are rigorously established. Then, it is proved that the proposed scheme is unconditionally convergent in the discrete L2 and Hγ/2 norm, where γ is a fractional order. Furthermore, we demonstrate that when the fractional order γ and the spatial grid number J increase, the block-Toeplitz coefficient matrix generated by the spatial discretization becomes ill-conditioned. As a result, we adopt an effective linearized iteration method for the nonlinear system, allowing it to be solved efficiently by the Krylov subspace solver with an appropriate circulant preconditioner, in which the fast Fourier transform is applied to speed up the computational cost at each iterative step. Finally, numerical experiments are presented to validate the theoretical findings and the efficiency of the fast algorithm.

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 High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

A compact finite difference scheme with absorbing boundary condition for forced KdV equation

Studying the long-time solution behavior of the Korteweg-de Vries (KdV) type equation with a periodic force acting at one end of the long channel is important for simulating the blood flow in artery driven by heart pulses. It is of great interest to develop an accurate numerical method for solving the forced KdV problem. In this article, we present the following methods to obtain an accurate approximation to the solution of KdV problem.•An accurate compact finite difference scheme is proposed for solving the above forced KdV problem with fourth-order accuracy.•An absorbing boundary condition at the right end of the interval is used to avoid the wave reflection.•The stability of scheme is proved by the von Neumann method and then tested by three examples. Results show that the method provides an accurate solution, and the wave propagates without reflection.

3D time-dependent scattering about complex shapes using high order difference potentials

We compute the scattering of unsteady acoustic waves about complex three-dimensional bodies with high order accuracy. The geometry of a scattering body is defined with the help of CAD. Its surface is represented as a collection of non-overlapping patches, each parameterized independently by means of high order splines (NURBS). As a specific example, we consider a submarine-like scatterer constructed using five different patches.The acoustic wave equation on the region exterior to the scatterer is solved by first reducing it to a system of Calderon's boundary operator equations. The latter are obtained using the method of difference potentials coupled with a compact fourth order accurate finite difference scheme. When solving the boundary operator equations, we employ Huygens' principle. It allows us to work on a sliding time window of non-increasing duration rather than keep the entire temporal history of the solution at the boundary.The proposed methodology demonstrates grid-independent computational complexity at the boundary and sub-linear complexity with respect to the grid dimension. It efficiently handles complex non-conforming geometries on Cartesian grids with no penalty for either accuracy or stability due to the cut cells. Its performance does not deteriorate over arbitrarily long simulation times. The exact treatment of artificial outer boundaries is inherently built in. Finally, multiple similar problems can be solved efficiently at a low individual cost per problem. This is important when, for example, the boundary condition on the surface changes but the scattering body stays the same.

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.

Highly Accurate Compact Finite Difference Schemes for Two-Point Boundary Value Problems with Robin Boundary Conditions

In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. Six examples for testing the applicability and performance of the method are considered. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs.

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.

Reconstruction of local volatility surface from American options

Abstract In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method.

A High Accuracy Local One-Dimensional Explicit Compact Scheme for the 2D Acoustic Wave Equation

In this paper, we develop a highly accurate and efficient finite difference scheme for solving the two-dimensional (2D) wave equation. Based on the local one-dimensional (LOD) method and Padé difference approximation, a fourth-order accuracy explicit compact difference scheme is proposed. Then, the Fourier analysis method is used to analyze the stability of the scheme, which shows that the new scheme is conditionally stable and the Courant-Friedrichs-Lewy (CFL) condition is superior to most existing methods of equivalent order of accuracy in the literature. Finally, numerical experiments demonstrate the high accuracy, stability, and efficiency of the proposed method.