Local One-dimensional Method Research Articles

A Compact Split-step Finite Difference Method for Solving the Nonlinear Schrödinger Equation

The nonlinear Schrödinger equation arises from quantum mechanics and is extensively used in many fields of science and engineering. Thus, it is important to construct the high-order and stable numerical scheme of the Schrödinger equation. To solve the high-order and stable numerical solution of the nonlinear Schrödinger equation, the compact split-step finite difference method and the local one-dimensional method are combined in this paper. To attain high-order accuracy in time and space, the 4-order compact finite difference in space discretization is combined with the L-stable Simpson method in time discretization. Therefore, a scheme with 4-order accuracy in space and 3-order accuracy in time is obtained, and the stability of the scheme is analyzed. Finally, numerical results manifest that the devised scheme can supply accurate and stable results to the nonlinear Schrödinger equation.

High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

An accurate and efficient local one-dimensional method for the 3D acoustic wave equation

Abstract We establish an accurate and efficient scheme with four-order accuracy for solving three-dimensional (3D) acoustic wave equation. First, the local one-dimensional method is used to transfer the 3D wave equation into three one-dimensional wave equations. Then, a new scheme is obtained by the Padé formulas for computation of spatial second derivatives and the correction of the truncation error remainder for discretization of temporal second derivative. It is compact and can be solved directly by the Thomas algorithm. Subsequently, the Fourier analysis method and the Lax equivalence theorem are employed to prove the stability and convergence of the present scheme, which shows that it is conditionally stable and convergent, and the stability condition is superior to that of most existing numerical methods of equivalent order of accuracy in the literature. It allows us to reduce computational cost with relatively large time step lengths. Finally, numerical examples have demonstrated high accuracy, stability, and efficiency of our method.

A high-order compact LOD difference method for solving the two-dimensional diffusion reaction equation with nonlinear source term

In this paper, a high-order compact local one-dimensional (LOD) method for the two-dimensional diffusion reaction equation with nonlinear source term is proposed. Firstly, the original equation is split into two one-dimensional equations. Then, the fourth-order compact difference formulas and the Crank–Nicolson method are used to approximate the space and time derivatives, respectively. The nonlinear source term is linearized based on the Taylor expansions. So we only need to solve two one-dimensional tri-diagonal systems at each time level, which makes the computation efficient. Through theoretical analysis, we find that the current LOD method produces a splitting error. Therefore, although this method has advantages for source term symmetry problems, it loses accuracy for source term asymmetry problems. The convergence and stability of the proposed scheme are proved by energy analysis method. Finally, several numerical examples are given to verify the accuracy and effectiveness of the proposed method. Furthermore, it is also applied in simulation of the quenching and blow-up phenomena.

Pseudo-transient ghost fluid methods for the Poisson-Boltzmann equation with a two-component regularization

The Poisson Boltzmann equation (PBE) is a well-established implicit solvent continuum model for the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is still a challenge due to its exponential nonlinear term, strong singularity by the source terms, and distinct dielectric regions. In this paper, a new pseudo-transient approach is proposed, which combines an analytical treatment of singular charges in a two-component regularization, with an analytical integration of nonlinear term in pseudo-time solution. To ensure efficiency, both fully implicit alternating direction implicit (ADI) and unconditionally stable locally one-dimensional (LOD) methods have been constructed to decompose three-dimensional linear systems into one-dimensional (1D) ones in each pseudo-time step. Moreover, to accommodate the nonzero function and flux jumps across the dielectric interface, a modified ghost fluid method (GFM) has been introduced as a first order accurate sharp interface method in 1D style, which minimizes the information needed for the molecular surface. The 1D finite-difference matrix generated by the GFM is symmetric and diagonally dominant, so that the stability of ADI and LOD methods is boosted. The proposed pseudo-transient GFM schemes have been numerically validated by calculating solvation free energy, binding energy, and salt effect of various proteins. It has been found that with the augmentation of regularization and GFM interface treatment, the ADI method not only enhances the accuracy dramatically, but also improves the stability significantly. By using a large time increment, an efficient protein simulation can be realized in steady-state solutions. Therefore, the proposed GFM-ADI and GFM-LOD methods provide accurate, stable, and efficient tools for biomolecular simulations.

Efficient and accurate numerical methods for the multidimensional convection–diffusion equations

A class of high order compact methods combined with local one-dimensional method have been studied to numerically solve multidimensional convection–diffusion equations. The methods are widely accepted due to their compactness, high accuracy. In this kind of methods the spatial derivatives are approximated implicitly rather than explicitly with smaller stencil but with higher accuracy. The local one-dimensional strategy is adopted in time to reduce the scale of algebraic equations resulting from numerical methods. This makes the multidimensional problems be easily coded. Based on analyzing the splitting error of the local one-dimensional method, a more accurate scheme is obtained through minor modification on the original scheme. By Von Neumann approach, we can find that the proposed schemes are unconditionally stable. Some numerical results are reported to illustrate that the schemes are robust, efficient and accurate.

A Comparison Study of ADI and LOD Methods on Option Pricing Models

This paper will focus on pricing options in marketing with two basic assets with risk and one basic asset without risk. In so doing, the Black-Scholes model and the European options which is applicable at the due date were used. By investigating the European option to find the proper price, it is necessary to solve an equation with partial derivatives which has two spatial variables. The finite differences will be used for such equations. Finite differences for one dimensional equations commonly ends in a three diagonal set which will be solved by calculation costs O(n) in which n is the number of discrete points. But here, since the problems are two dimensional, the Alternating Direction Implicit (ADI) and Locally One-Dimensional (LOD) are used to reduce the calculation costs. The open cost is at the level of discrete points and this is the advantage of these methods. Moreover, these methods enjoy acceptable stability. Though ADI and LOD are equal and easy in calculations, evaluating these methods in pricing the option indicates that the ADI method is sensitive to discontinuity or non-derivation which is the common property of income function; therefore, this thesis proposes the LOD method.

Comprehensive Numerical Analysis of Finite Difference Time Domain Methods for Improving Optical Waveguide Sensor Accuracy

This paper discusses numerical analysis methods for different geometrical features that have limited interval values for typically used sensor wavelengths. Compared with existing Finite Difference Time Domain (FDTD) methods, the alternating direction implicit (ADI)-FDTD method reduces the number of sub-steps by a factor of two to three, which represents a 33% time savings in each single run. The local one-dimensional (LOD)-FDTD method has similar numerical equation properties, which should be calculated as in the previous method. Generally, a small number of arithmetic processes, which result in a shorter simulation time, are desired. The alternating direction implicit technique can be considered a significant step forward for improving the efficiency of unconditionally stable FDTD schemes. This comparative study shows that the local one-dimensional method had minimum relative error ranges of less than 40% for analytical frequencies above 42.85 GHz, and the same accuracy was generated by both methods.

On the positivity, monotonicity, and stability of a semi-adaptive LOD method for solving three-dimensional degenerate Kawarada equations

This paper concerns the numerical solution of three-dimensional degenerate Kawarada equations. These partial differential equations possess highly nonlinear source terms, and exhibit strong quenching singularities which pose severe challenges to the design and analysis of highly reliable schemes. Arbitrary fixed nonuniform spatial grids, which are not necessarily symmetric, are considered throughout this study. The numerical solution is advanced through a semi-adaptive Local One-Dimensional (LOD) integrator. The temporal adaptation is achieved via a suitable arc-length monitoring mechanism. Criteria for preserving the positivity and monotonicity are investigated and acquired. The numerical stability of the splitting method is proven in the von Neumann sense under the spectral norm. Extended stability expectations are proposed and investigated.

A compact LOD method and its extrapolation for two-dimensional modified anomalous fractional sub-diffusion equations

A Crank–Nicolson-type compact locally one-dimensional (LOD) finite difference method is proposed for a class of two-dimensional modified anomalous fractional sub-diffusion equations with two time Riemann–Liouville fractional derivatives of orders (1−α) and (1−β)(0<α,β<1). The resulting scheme consists of simple tridiagonal systems and all computations are carried out completely in one spatial direction as for one-dimensional problems. This property evidently enhances the simplicity of programming and makes the computations more easy. The unconditional stability and convergence of the scheme are rigorously proved. The error estimates in the standard H1- and L2-norms and the weighted L∞-norm are obtained and show that the proposed compact LOD method has the accuracy of the order 2min{α,β} in time and 4 in space. A Richardson extrapolation algorithm is presented to increase the temporal accuracy to the order min{α+β,4min{α,β}} if α≠β and min{1+α,4α} if α=β. A comparison study of the compact LOD method with the other existing methods is given to show its superiority. Numerical results confirm our theoretical analysis, and demonstrate the accuracy and the effectiveness of the compact LOD method and the extrapolation algorithm.

A compact locally one-dimensional method for fractional diffusion-wave equations

This paper is concerned with numerical methods for a class of multi-dimensional fractional diffusion-wave equations with a time fractional derivative of order $$\alpha $$ ( $$1<\alpha <2$$ ). A compact locally one-dimensional (LOD) finite difference method is proposed for the equations. The resulting scheme consists of one-dimensional tridiagonal systems, and all computations are carried out completely in one spatial direction as for one-dimensional problems. The unconditional stability and $$H^{1}$$ norm convergence of the scheme are rigorously proved for the three-dimensional case. The error estimates show that the proposed compact LOD method converges with the order $$(3-\alpha )$$ in time and $$4$$ in space. Numerical results confirm our theoretical analysis and illustrate the effectiveness of this new method.

A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations

A higher-order compact locally one-dimensional (LOD) finite difference method for two-dimensional nonhomogeneous parabolic differential equations is proposed. The resulting scheme consists of two one-dimensional tridiagonal systems, and all computations are implemented completely in one spatial direction as for one-dimensional problems. The solvability and the stability of the scheme are proved almost unconditionally. The error estimates are obtained in the discrete H1,L2 and L∞ norms, and show that the proposed compact LOD method has the accuracy of the second-order in time and the fourth-order in space. Two Richardson extrapolation algorithms are presented to increase the accuracy to the fourth-order and the sixth-order in both time and space when the time step is proportional to the spatial mesh size. Numerical results demonstrate the accuracy of the compact LOD method and the high efficiency of its extrapolation algorithms.

Fundamental Locally One-Dimensional Method for 3-D Thermal Simulation

This paper presents a fundamental locally one-dimensional (FLOD) method for 3-D thermal simulation. We first propose a locally one-dimensional (LOD) method for heat transfer equation within general inhomogeneous media. The proposed LOD method is then cast into compact form and formulated into the FLOD method with operator-free right-hand-side (RHS), which leads to computationally efficient update equations. Memory storage requirements and boundary conditions for both FLOD and LOD methods are detailed and compared. Stability analysis by means of analyzing the eigenvalues of amplification matrix substantiates the stability of the FLOD method. Additionally, the potential instability of the Douglas Gunn (DG) alternating-direction-implicit (ADI) method for inhomogeneous media is demonstrated. Numerical experiments justify the gain achieved in the overall efficiency for FLOD over LOD, DG-ADI and explicit methods. Furthermore, the relative maximum error of the FLOD method illustrates good trade-off between accuracy and efficiency.

An Unconditionally Stable 1-D FDTD Algorithm for Modeling Chiral Media Based on Similar LOD Method

An unconditionally stable scheme, which employs a similar technique to the locally one-dimensional (LOD) method, is proposed to study the one-dimensional (1D) nondispersive chiral media. In the proposed scheme, the recent new mesh-dividing method for bi-isotropic media is used. The method regards the two terms on the right-hand side of the rearranged curl equations as two directions and uses the LOD algorithm to deal with the equivalent two-dimensional (2D) problem. Through Fourier stability analysis and numerical simulation, it is found that the time-step size is not restricted by the Courant-Friedrich-Levy (CFL) stability condition, hence the proposed scheme is unconditionally stable. Compared with the similar Alternating Direction Implicit (ADI) method, the proposed scheme has lighter calculation burden and higher accuracy.

Error and extrapolation of a compact LOD method for parabolic differential equations

This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.

A Numerical Solution of the Two-Dimensional Fusion Problem with Convective Boundary Conditions

In this paper, we present an LOD method for solving the two-dimensional fusion problem with convective boundary conditions. In this study, we extend our earlier work [1] on the solution of the two-dimensional fusion problem by considering a class of time-split finite-difference methods, namely locally one-dimensional (LOD) schemes. In addition, following the idea of Douglas [2, 3], a Douglas-like splitting scheme is presented. A stability analysis by Fourier series method (von Neumann stability) of the scheme is also investigated. Computational results obtained by the present method are in excellent agreement with the results reported previously by other research.

High-order schemes for acoustic waveform simulation

This article introduces a new fourth-order implicit time-stepping scheme for the numerical solution of the acoustic wave equation, as a variant of the conventional modified equation method. For an efficient simulation, the scheme incorporates a locally one-dimensional (LOD) procedure having the splitting error of O ( Δ t 4 ) . Its stability and accuracy are compared with those of the standard explicit fourth-order scheme. It has been observed from various experiments for 2D problems that (a) the computational cost of the implicit LOD algorithm is only about 40% higher than that of the explicit method, for the problems of the same size, (b) the implicit LOD method produces less dispersive solutions in heterogeneous media, and (c) its numerical stability and accuracy match well those of the explicit method.

ON A LOD METHOD FOR SOLUTION OF TWO DIMENSIONAL FUSION PROBLEM WITH CONVECTIVE BOUNDARY CONDITIONS

Finite-difference technique based on explicit method for one-dimensional fusion are used to solve the two-dimensional time dependent fusion equation with convective boundary conditions. In these cases locally one-dimensional (LOD) techniques is used to extend the one-dimensional problem

IMPROVED ACCURACY FOR LOCALLY ONE-DIMENSIONAL METHODS FOR PARABOLIC EQUATIONS

Classical alternating direction (AD) and fractional step (FS) methods for parabolic equations, based on some standard implicit time-stepping procedure such as Crank–Nicolson, can have errors associated with the AD or FS perturbations that are much larger than the errors associated with the underlying time-stepping procedure. We show that minor modifications in the AD and FS procedures can virtually eliminate the perturbation errors at an additional computational cost that is less than 10% of the cost of the original AD or FS method. Moreover, after these modifications, the AD and FS procedures produce identical approximations of the solution of the differential problem. It is also shown that the same perturbation of the Crank–Nicolson procedure can be obtained with AD and FS methods associated with the backward Euler time-stepping scheme. An application of the same concept is presented for second-order wave equations.

New explicit finite difference schemes for two-dimensional diffusion subject to specification of mass

Two different explicit finite difference schemes for the numerical solution of the diffusion equation on a rectangular region, subject to local or nonlocal boundary conditions, the latter involving a double integral to simulate specification of mass in a curved region, are compared. These schemes, the two-dimensional 9-point Forward Time Centered Space (FTCS) explicit formula [Noye & Hayman, J Comp Math 42, 1992, 223–236] and the locally one-dimensional (LOD) method based on the classical one-dimensional FTCS formula [Noye & Hayman, J Comp Math 51, 1994, 215–228], are economical to use, are generally second-order, have bounded ranges of stability, and can be shown to be identical at grid points in the interior of the solution domain. However, results obtained are different, unless a special boundary treatment is used with the LOD method. Then the LOD method is more efficient. Some numerical tests are presented for both cases, and accuracy and Central Processor (CP) time needed for the nonlocal problem are found to be superior than those for the method of Cannon et al. [Cannon et al., Appl Anal J 50, 1993, 1–19]. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 521–534, 1999