Second-order Convergence Rate Research Articles

Efficient second-order accurate exponential time differencing for time-fractional advection-diffusion-reaction equations with variable coefficients

Time-fractional advection-diffusion-reaction type equations are useful for characterizing anomalous transport processes. In this paper, linearly implicit as well as explicit generalized exponential time differencing (GETD) schemes are proposed for solving a class of such equations having time-space dependent coefficients. The implicit scheme, being unconditionally stable, is robust in handling the numerical instabilities in problems where the advection term is dominant. Regarding the error analysis, uniformly optimal second-order convergence rates are derived using time-graded meshes to counter the effect of the inherent singularity of the continuous solution. Implementation of generalized exponential integrators requires computing the action of Mittag-Leffler function of matrices on a vector, or on a matrix in the case of the implicit scheme. For cost-effective implementation, using global Padé approximants these computation tasks get reduced to solving linear systems. A new approach based on Sylvester equation formulation of the resulting linear systems is developed in this paper. This technique leads to significantly faster algorithms for implementing the GETD schemes. Numerical experiments are provided to illustrate the theoretical findings and to assert the efficiency of the Sylvester equation based approach. Application of this approach to an existing GETD scheme for solving a nonlinear subdiffusion problem is also discussed.

Second-order error analysis of a corrected average finite difference scheme for time-fractional Cable equations with nonsmooth solutions

In this paper, we first formulate an average finite difference scheme with appropriate correction terms for the time-fractional initial-value problem of y′(t)+0RLDt1−αy(t)=g(t), where 0RLDt1−α denotes the Riemann–Liouville fractional derivative of order 1−α(α∈(0,1)). It is shown that, under some suitable conditions on the data, the numerical solution converges to the exact solution with order O(τ2), where τ is the time step size. The method is then extended to solve the time-fractional Cable equation, combined with a standard discretisation of the spatial derivatives on a uniform mesh. The second-order time convergence rate is proved. Several numerical examples are given to illustrate the good agreement with the theoretical analysis of the presented methods.

Macroscopic finite-difference scheme and modified equationsof the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations(MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation(NACDE) and Navier-Stokes equations(NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation(CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O(τ2) in H1- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

A Non-Traditional Finite Element Method for Scattering by Partly Covered Grooves with Multiple Media

We present a non-traditional finite element method for the electromagnetic scattering of the partly covered groove filled with multiple media. The non-local boundary condition is introduced to make the unbounded scattering into a bounded domain problem. Non-body-fitted mesh is applied to save the cost of discretizing groove domain greatly. In addition, the level set functions are utilized to describe complex media interfaces and boundaries. Numerical tests demonstrate the effectiveness and second-order convergence rate of the proposed method.

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

A monotone finite volume scheme for single phase flow with reactive transport in anisotropic porous media

This paper deals with development and analysis of a finite volume scheme for a nonlinear parabolic equation modeling a single phase flow with reactive solute transport, including equilibrium sorption, in heterogeneous porous media. We present a second-order, finite volume scheme by combining a nonlinear two point flux approximation for the diffusion term, and an upwind method for the advective term on unstructured meshes. It is shown that, under a CFL condition, this scheme ensures the non-negativity of the discrete solution, taking into account the anisotropy and heterogeneity of the domain. The existence of a solution for the discrete nonlinear system is studied, and the convergence of a fixed point method is proved. Our theoretical results are illustrated by numerical simulations in a bidimensional domain, in particular we show the second-order convergence rate for the numerical solution and confirm the monotonicity of the scheme.

A robust higher-order fitted mesh numerical method for solving singularly perturbed parabolic reaction-diffusion problems

In this study, a robust higher-order numerical method for solving singularly perturbed parabolic reaction-diffusion problems is presented. The Crank-Nicolson method is applied to discretize the time derivative on a uniform mesh. On a Shishkin mesh, the space derivative is discretized using a hybrid numerical method that combines the cubic spline in tension method for the boundary layer regions with the central difference method for the outer region. Theoretically, we proved that the proposed hybrid numerical method is second-order in the outer region and fourth-order in the boundary layer regions in the space direction. As a result of this, the proposed method produces an almost second-order rate of convergence in the time domain and a higher-order rate of convergence in the space domain. The newly developed method is numerically demonstrated to be uniformly convergent at higher-order, independent of the perturbation parameter. Three numerical examples are computed to support the theoretical results.

The splitting-based semi-implicit finite-difference schemes for simulation of blood flow in arteries

This paper is devoted to construction and analysis of splitting-based finite-difference schemes for simulation of blood flow in applications of one-dimensional models. Proposed schemes are constructed for inviscid and viscid models of blood. The proposed approach is based on the modification of averaged incompressibility condition, rewritten for the square root of a cross-sectional area. It is approximated by an absolutely stable scheme with a skew-symmetric operator. The constructed schemes can be classified as semi-implicit. The stability of proposed schemes is analyzed by the method of energy inequalities. Sufficient stability conditions are obtained. For the problems with analytical solutions, it is demonstrated that a second-order convergence rate is achieved in practice.The proposed schemes are compared with the widely used explicit finite-difference schemes, such as Lax–Wendroff, Lax–Friedrichs and McCormack schemes. The benchmark problems for simulation of flows in the following vascular systems are considered: human carotid artery, human aorta with bifurcation, vessel with bifurcation and absorbing outflow conditions, and networks with 31, 63, and 127 vessels. It is demonstrated that the computations based on the splitting-based semi-implicit schemes can be performed much faster than for the explicit schemes with close values of relative errors.

A highly efficient semi-implicit corrective SPH scheme for 2D/3D tumor growth model

In this work, a semi-implicit corrective SPH method is proposed to solve the multi-component Cahn–Hilliard equation with fourth-order derivatives, and it is further used to predict the high-dimensional tumor growth model. The scheme is motivated by: (a) the fourth-order spatial derivative is discretized continuously by a corrective SPH formula for approximating second-order derivative twice, and the Neumann boundary is imposed by a ghost technique; (b) the temporal direction is approximated by the implicit scheme, and an iterative concept is employed to handle the above implicit form; (c) the multi-CPUs MPI parallelization is adopted to reduce the computing cost. Firstly, the second-order convergence rate of the proposed method for 2D/3D equation is shown and discussed by two analytical examples, and the mass conservation and energy properties are also demonstrated. Secondly, the efficiency of the proposed approach for multi-phase separation phenomenon is illustrated in an irregular domain. Finally, the 2D/3D tumor growth evolution at a short time is predicted by the proposed method and qualitatively compared with other numerical results. The numerical experiments show that the proposed scheme for phase-separation phenomenon or tumor growth model is highly efficient and reliable.

The local discontinuous Galerkin method for the nonlinear quantum Zakharov system

In this paper, we design an energy conserving local discontinuous Galerkin (LDG) method to solve the quantum Zakharov system (QZS). We first give theoretical proof for the conservation of mass and energy and then focus our discussion on the priori estimates for the semi-discrete approximation of the QZS. With the proof for the L2 boundedness of the numerical solutions and the discrete Poincaré inequalities, we obtain the optimal error estimates. Next, we develop a decoupled and implicit linear fully-discrete scheme using the semi-implicit spectral deferred correction (SISDC) method for temporal discretization. Another strength of our approach is the arbitrary high accuracy which depends only on the degree of the approximating piecewise polynomials. Finally, several numerical examples are reported to validate it and the conservation properties. Especially, the second-order convergence rate of the QZS to its limiting model in the semi-classical limit is achieved as expected.

An arbitrary Lagrangian–Eulerian method for analyzing finite-amplitude viscous acoustic waves radiated from vibrational solid boundaries: An implicit method

This paper is concerned with the development of an arbitrary Lagrangian–Eulerian (ALE) method for predicting nonlinear acoustic waves radiated from large-amplitude vibrational solid boundaries immersed in viscous fluids of infinite extent. The method is formulated based on a nonlinear acoustic model characterized by velocity potential and acoustic pressure. Unsplit implementations of three perfectly matched layer (PML) techniques are presented, namely conventional PML, convolution PML, and complex frequency-shifted PML (CFS-PML), to enable bounded-domain modeling of nonlinear acoustic waves propagating in an unbounded fluid medium. The long-time stability characteristics of the three distinct PML techniques are examined through theoretical analysis and numerical verification of their ability to absorb evanescent waves. Our results reveal that the CFS-PML technique exhibits superior long-time stability performance compared to the other two techniques. The CFS-PML is also compared to two different absorbing layer techniques: the Kosloff Absorbing Layer (KAL) technique and the Acoustic Black Hole (ABH) technique. The finite element method is adopted for spatial discretization of the finite-amplitude acoustic wave model, and an implicit predictor–corrector algorithm is proposed for time advancement. Several benchmark problems, including acoustic waves produced by a vibrational piston, and transversely oscillating and pulsating cylinders, are analyzed by the proposed method. It is found that the proposed method attains a second-order rate of convergence for spatial discretization, and the convergence rate in terms of the time step size is about first-order. The application of the method to acoustic levitation problems is also demonstrated, and the nonlinear phenomenon of acoustic waves in a single-axis acoustic levitator is discussed.

An arbitrarily Lagrangian–Eulerian SPH scheme with implicit iterative particle shifting procedure

This work presents an Arbitrarily Lagrangian–Eulerian Smoothed Particle Hydrodynamics (ALE-SPH) scheme for Navier–Stokes equations where the particle distribution is optimized by an Implicit Iterative Particle Shifting (IIPS) correction capable of improving the accuracy of the SPH spatial operators. Here two different possible approaches, based respectively on Moving Least Square (MLS) and on convective fluxes have been adopted to embed the IIPS formulation in an ALE-SPH scheme. The theoretical second-order convergence rate has been achieved for the Taylor–Green flow at Reynolds number Re=100. Results obtained with the proposed numerical scheme for the Moving box and the Impinging jet test cases confirm that the accuracy is significantly improved, with respect to existing quasi-Lagrangian ALE-SPH schemes.

Full waveform inversion based on hybrid conjugate gradient with BFGS direction

Full waveform inversion (FWI) has the problem of low computational efficiency. The main reasons are low convergence rate and unstable convergence. We mainly use optimization methods to solve computationally inefficient problems. Among the commonly used optimization methods, the L-BFGS (Limited memory Broyden-Fletcher-Goldfarb-Shanno) method converges fast but not stably, while the HCG (Hybrid Conjugate gradient) method converges stably but slowly. Neither of these methods can maximize the computational efficiency of FWI. By using Gauss-Newton direction to approximate the search direction (QN-HCG) of HCG method, the second-order convergence rate of Gauss-Newton method and the strong convergence stability of HCG method can be used at the same time. However, QN-HCG requires the Hessian matrix to be positive definite, which is complicated to compute and occupies a large memory. The memoryless BFGS method is developed from the L-BFGS method. Since the Hessian matrix is not required to be positive definite, the method is simple to compute and occupies less memory, and is therefore more suitable to approximate the search direction of the HCG method. First, in order to improve the convergence speed of FWI and ensure the stability of convergence, we use BFGS without memory variable to approximate the search direction of HCG method (BFGS-HCG), and introduce this method into FWI. Second, the computational efficiency of the proposed method is verified in model tests of acoustic and elastic wave FWI. Model tests show that this approach can improve the computational efficiency of full waveform inversion.

Second-order numerical method for coupling of slightly compressible Brinkman flow with advection-diffusion system in fractured media

This paper considers a coupling of slightly compressible Darcy-Brinkman-transport problem in fractured media with higher Reynolds numbers, involving Brinkman flow in the fractures with advection-diffusion transport in the whole considered media. A new two-layer reduced coupled model is introduced by treating the fracture as hyperplane. The finite difference method with second-order backward differentiation formula and modified upwind scheme by using the fewest nodal points is constructed to solve the new reduced coupled model on staggered nonuniform grids. Based on error derivation of the coupling term, the uniqueness and existence of solutions and second-order convergence rate of numerical method are derived in the mixed forms under the mass conservative transmission and Beavers-Joseph-Saffman conditions. Some experiments are provided to testify the accuracy and efficiency of the numerical method. The numerical analysis of new two-layer reduced model is illustrated to show the behavior of fluid flow and solute transport in the media with intersected, embedded and L-shaped fractures.

Uniformly convergent hybrid numerical scheme for singularly perturbed turning point problems with delay

In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε.

A two-grid ADI finite element approximation for a nonlinear distributed-order fractional sub-diffusion equation

<abstract><p>In this paper, a two-grid alternating direction implicit (ADI) finite element (FE) method based on the weighted and shifted Grünwald difference (WSGD) operator is proposed for solving a two-dimensional nonlinear time distributed-order fractional sub-diffusion equation. The stability and optimal error estimates with second-order convergence rate in spatial direction are obtained. The storage space can be reduced and computing efficiency can be improved in this method. Two numerical examples are provided to verify the theoretical results.</p></abstract>

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.