Space Step Size Research Articles

An energy stable, hexagonal finite difference scheme for the 2D phase field crystal amplitude equations

In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.

Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations

Using finite element method in spatial direction and classical L1 approximation in temporal direction, a fully-discrete scheme is established for a class of two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivatives. The stability analysis of the approximate scheme is proposed. The spatial global superconvergence and temporal convergence of order O(h2+τ2−α) for the original variable in H1-norm is presented by means of properties of bilinear element and interpolation postprocessing technique, where h and τ are the step sizes in space and time, respectively. Finally, several numerical examples are implemented to evaluate the efficiency of the theoretical results.

Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions

AbstractThis article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with convergence in the L1(L∞)-norm for the one-dimensional case, where τ,h and σ are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and convergent in the L1(L∞)-norm, where h1 and h2 are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.

Improved efficient difference method for the modified anomalous sub-diffusion equation with a nonlinear source term

ABSTRACTIn this paper, we propose an improved efficient difference method to solve the modified anomalous sub-diffusion equation with a nonlinear source term. The weighted and shifted Grünwald difference operator and compact difference are used to approximate the Riemann–Liouville fractional derivative and the space derivative, respectively. Since the semi-discrete system obtained by discretizing in space has some stiff properties when the space mesh is dense, the second-order backward differentiation formula (BDF) is then employed to solve the semi-discrete system and a second-order interpolation formula is used to deal with the nonlinear source term. The stability and convergence of the method are discussed and the convergence order is shown to be , where τ and h denote the time and space step size. The new method has higher accuracy compared with the existed methods and good stability in the numerical performance. Numerical examples are given to illustrate the theoretical results.

Quadratic spline collocation method for the time fractional subdiffusion equation

In this paper, exploiting the quadratic spline collocation (QSC) method, we numerically solve the time fractional subdiffusion equation with Dirichelt boundary value conditions. The coefficient matrix of the discretized linear system is investigated in detail. Theoretical analyses and numerical examples demonstrate the proposed technique can enjoy the global error bound with O(τ3+h3) under the L∞ norm provided that the solution v(x, t) has four-order continual derivative with respects to x and t, and it can achieve the accuracy of O(τ4+h4) at collocation points, where τ, h are the step sizes in time and space, respectively.

A neural network-based 2D/3D image registration quality evaluator for pediatric patient setup in external beam radiotherapy.

Our purpose was to develop a neural network‐based registration quality evaluator (RQE) that can improve the 2D/3D image registration robustness for pediatric patient setup in external beam radiotherapy. Orthogonal daily setup X‐ray images of six pediatric patients with brain tumors receiving proton therapy treatments were retrospectively registered with their treatment planning computed tomography (CT) images. A neural network‐based pattern classifier was used to determine whether a registration solution was successful based on geometric features of the similarity measure values near the point‐of‐solution. Supervised training and test datasets were generated by rigidly registering a pair of orthogonal daily setup X‐ray images to the treatment planning CT. The best solution for each registration task was selected from 50 optimizing attempts that differed only by the randomly generated initial transformation parameters. The distance from each individual solution to the best solution in the normalized parametrical space was compared to a user‐defined error tolerance to determine whether that solution was acceptable. A supervised training was then used to train the RQE. Performance of the RQE was evaluated using test dataset consisting of registration results that were not used in training. The RQE was integrated with our in‐house 2D/3D registration system and its performance was evaluated using the same patient dataset. With an optimized sampling step size (i.e., 5 mm) in the feature space, the RQE has the sensitivity and the specificity in the ranges of 0.865–0.964 and 0.797–0.990, respectively, when used to detect registration error with mean voxel displacement (MVD) greater than 1 mm. The trial‐to‐acceptance ratio of the integrated 2D/3D registration system, for all patients, is equal to 1.48. The final acceptance ratio is 92.4%. The proposed RQE can potentially be used in a 2D/3D rigid image registration system to improve the overall robustness by rejecting unsuccessful registration solutions. The RQE is not patient‐specific, so a single RQE can be constructed and used for a particular application (e.g., the registration for images acquired on the same anatomical site). Implementation of the RQE in a 2D/3D registration system is clinically feasible.PACS numbers: 87.57.nj, 87.85.dq, 87.55.Qr

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order $$0<\alpha <1$$0<ź<1. As to the conforming scheme, the spatial global superconvergence and temporal convergence order of $$O(h^2+\tau ^{2-\alpha })$$O(h2+ź2-ź) for both the original variable u in $$H^1$$H1-norm and the flux $$\vec {p}=\nabla u$$pź=źu in $$L^2$$L2-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and $$\tau $$ź are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of $$\textit{EQ}_1^{\textit{rot}}$$EQ1rot nonconforming element, which manifests that convergence orders of $$O(h+\tau ^{2-\alpha })$$O(h+ź2-ź) and $$O(h^2+\tau ^{2-\alpha })$$O(h2+ź2-ź) for the original variable u in broken $$H^1$$H1-norm and $$L^2$$L2-norm, respectively, and approximation for the flux $$\vec {p}$$pź converging with order $$O(h+\tau ^{2-\alpha })$$O(h+ź2-ź) in $$L^2$$L2-norm. Numerical examples are provided to demonstrate the theoretical analysis.

Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation

In this paper we devise and analyze an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in $L^\infty (0,T;L^\infty)$ and the discrete chemical potential is bounded in $L^\infty (0,T;L^2)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme.

Efficiency-Improved 3-D WCS-FDTD Algorithm for Periodic Structures at Oblique Incidence

For solving the problems of oblique incident wave on periodic structures, an efficiency-improved 3-D weakly conditionally stable finite-difference time-domain (WCS-FDTD) method is presented in this letter, whose Courant–Friedrichs–Lewy (CFL) stability condition is only determined by one space step size. By expressing the extra terms caused by the phase delay as the form of the coefficient matrix, and combining with the locally one-dimensional (LOD) scheme and the inverse matrix operator, the Maxwell’s equations are split into two sub-step procedures. Further, by introducing a set of auxiliary field variables denoted by ${\psi}$ , the proposed algorithm is reformulated in a much simpler form with more concise right-hand sides for an efficient implementation. Numerical example demonstrates that the proposed method is more efficient while keeping the same CFL stability condition and memory storage requirement as that of the existing method.

Structural Mapping of Functional Ge Layers Grown on Graded SiGe Buffers for sub-10 nm CMOS Applications Using Advanced X-ray Nanodiffraction.

We report a detailed advanced materials characterization study on 40 nm thick strained germanium (Ge) layers integrated on 300 mm Si(001) wafers via strain-relaxed silicon-germanium (SiGe) buffer layers. Fast-scanning X-ray microscopy is used to directly image structural inhomogeneities, lattice tilt, thickness, and strain of a functional Ge layer down to the sub-micrometer scale with a real space step size of 750 μm. The structural study shows that the metastable Ge layer, pseudomorphically grown on Si(0.3)Ge(0.7), exhibits an average compressive biaxial strain of -1.27%. By applying a scan area of 100 × 100 μm(2), we observe microfluctuations of strain, lattice tilt, and thickness of ca. ±0.03%, ±0.05°, and ±0.8 nm, respectively. This study confirms the high materials homogeneity of the compressively strained Ge layer realized by the step-graded SiGe buffer approach on 300 mm Si wafers. This presents thus a promising materials science approach for advanced sub-10 nm complementary metal oxide-semiconductor applications based on strain-engineered Ge transistors to outperform current Si channel technologies.

Finite element method for space-time fractional diffusion equation

In this paper, we consider two types of space-time fractional diffusion equations(STFDE) on a finite domain. The equation can be obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann-Liouville fractional derivative of order s (1 <s ≤ 2), and the first order time derivative by a Caputo fractional derivative of order ? (0 <? ≤ 1). For the 0 <? < 1 case, we present two schemes to approximate the time derivative and finite element methods for the space derivative, the optimal convergence rate can be reached O(?2?? + h2) and O(?2 + h2), respectively, in which ? is the time step size and h is the space step size. And for the case ? = 1, we use the Crank-Nicolson scheme to approximate the time derivative and obtain the optimal convergence rate O(?2 + h2) as well. Some numerical examples are given and the numerical results are in good agreement with the theoretical analysis.

An Algorithm for the Numerical Solution of Two-Sided Space-Fractional Partial Differential Equations

Abstract We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order O ( Δ x 3 - α ) ${O(\Delta x^{3- \alpha })}$ , 1 &lt; α &lt; 2 ${1&amp;lt;\alpha &amp;lt;2}$ . A shifted implicit finite difference method is applied for solving the two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is O ( Δ t + Δ x min ( 3 - α , β ) ) ${O (\Delta t + \Delta x^{\min (3- \alpha , \beta )})}$ , 1 &lt; α &lt; 2 ${1&amp;lt; \alpha &amp;lt;2}$ , β &gt; 0 ${\beta &amp;gt;0}$ , where Δ t , Δ x ${\Delta t,\Delta x}$ denote the time and space stepsizes, respectively. Numerical examples where the solutions have varying degrees of smoothness are presented and compared with the exact analytical solution to compare the practical performance of the method with the theoretical order of convergence.

High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations

In this paper, a high order approximation with convergence order $$O( {{\tau ^{3 - \alpha }}})$$ to Caputo derivative $$_{C}D_{0,t}^{\alpha}f(t)$$ for $$\alpha\in(0,1)$$ is introduced. Furthermore, two high-order algorithms for Caputo type advection-diffusion equation are obtained. The stability and convergence are rigorously studied which depend upon the derivative order \alpha. The corresponding convergence orders are $$O(\tau^{3-\alpha}+h^2)$$ and $$O(\tau^{3-\alpha}+h^4)$$ where \tau is the time stepsize, h the space stepsize, respectively. Finally, numerical examples are given to support the theoretical analysis.

Some high-order difference schemes for the distributed-order differential equations

Two difference schemes are derived for both one-dimensional and two-dimensional distributed-order differential equations. It is proved that the schemes are unconditionally stable and convergent in an L1(L∞) norm with the convergence orders O(τ2+h2+Δα2) and O(τ2+h4+Δα4), respectively, where τ, h and Δα are the step sizes in time, space and distributed-order variables. Several numerical examples are given to confirm the theoretical results.

Fast weak–KAM integrators for separable Hamiltonian systems

We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a {\em geometric integrator} satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.

Devising efficient numerical methods for oscillating patterns in reaction–diffusion systems

In this paper, we consider the numerical approximation of a reaction–diffusion system 2D in space whose solutions are patterns oscillating in time or both in time and space. We present a stability analysis for a linear test heat equation in terms of the diffusion d and of the reaction timescales given by the real and imaginary parts α and β of the eigenvalues of J(Pe), the Jacobian of the reaction part at the equilibrium point Pe. Focusing on the case α=0,β≠0, we obtain stability regions in the plane (ξ,ν), where ξ=λ(h;d)ht, ν=βht, ht time stepsize, λ lumped diffusion scale depending also from the space stepsize h and from the spectral properties of the discrete Laplace operator arising from the semi-discretization in space. In space we apply the Extended Central Difference Formulas (ECDFs) of order p=2,4,6. In time we approximate the diffusion part in implicit way and the reaction part by a selection of integrators: the Explicit Euler and ADI methods, the symplectic Euler and a partitioned Runge–Kutta method that are symplectic in the absence of diffusion. Hence, by estimating λ, for each method we derive stepsize restrictions ht⪅Fmet(h;d,β,p) in terms of the stability curve Fmet depending on diffusion and reaction timescales and from the approximation order in space. For the same schemes, we provide also a dispersion error analysis. We present numerical simulations for the test heat equation and for the Lotka–Volterra PDE system with solutions oscillating only in time for the presence of a centre-type dynamics. In these cases, the implicit-symplectic schemes provide the best choice. We solve also the Schnakenberg model with spatial patterns oscillating in space and time in the presence of an attractive limit cycle due to the Turing–Hopf instability. In this case, all schemes attain closed orbits in the phase space, but the Explicit ADI method is the best choice from the computational point of view.

Wave propagation in muddy sediments using time domain finite difference approach

There is a renewed interest in understanding wave propagation in muddy sediments. In this study, we are using the time domain finite difference approach to investigate wave propagation in a muddy environment representative of the New England Mud Patch south of Martha’s Vineyard. The model uses the two-dimensional full-wave time-domain finite-difference code developed at WHOI over the past 35 years. In order to simplify the computation we consider perfectly elastic, isotropic media with uniform step sizes in time and space. The source waveform is a Ricker wavelet with a peak frequency in pressure of 200 Hz. This study considers a 100 m water waveguide (Vp = 1500 m/s) overlying 12 m of mud (Vp = 1530 m/s, Vs = 100 m/s) above a faster bottom (Vp = 1700 m/s, Vs = 500 m/s). The goal of the modeling was to understand the propagation of compressional, shear and interface waves in the presence of a soft muddy layer. [Work supported by ONR Code 3220A.]

Analysis of a Mixed Finite Element Method for a Cahn--Hilliard--Darcy--Stokes System

In this paper we devise and analyze a mixed finite element method for a modified Cahn--Hilliard equation coupled with a nonsteady Darcy--Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in $L^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential is bounded in $L^\infty \left(0,T;L^2\right)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions.

Numerical Algorithms for Time-Fractional Subdiffusion Equation with Second-Order Accuracy

This article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed for the time-fractional subdiffusion equation with space discretized by finite element method and time discretized by the fractional linear multistep methods. These two methods are unconditionally stable with maximum global convergence order of $O(\tau+h^{r+1})$ in the $L^2$ norm, where $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the piecewise polynomial space. The average convergence rates for the two methods in time are also investigated, which shows that the average convergence rates of the two methods are $O(\tau^{1.5}+h^{r+1})$. Furthermore, two improved algorithms are constrcted, and they are also unconditionally stable and convergent of order $O(\tau^2+h^{r+1})$. Numerical examples are provided to verify the theoretical analysis. Comparisons between the present algorithms and the existing ones are included, showing that our numerical algorithms exhibit better performances than the known ones.

On asymptotic global error estimation and control of finite difference solutions for semilinear parabolic equations

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.