Order Numerical Scheme Research Articles

A Third Order BDF Energy Stable Linear Scheme for the No-Slope-Selection Thin Film Model

In this paper we propose and analyze a (temporally) third order accurate backward differentiation formula (BDF) numerical scheme for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral discretization in space. The surface diffusion term is treated implicitly, while the nonlinear chemical potential is approximated by a third order explicit extrapolation formula for the sake of solvability. In addition, a third order accurate Douglas-Dupont regularization term, in the form of $-A \Delta t^2 \Delta_N^2 ( u^{n+1} - u^n)$, is added in the numerical scheme. A careful energy stability estimate, combined with Fourier eigenvalue analysis, results in the energy stability in a modified version, and a theoretical justification of the coefficient $A$ becomes available. As a result of this energy stability analysis, a uniform in time bound of the numerical energy is obtained. And also, the optimal rate convergence analysis and error estimate are derived in details, in the $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_h^2)$ norm, with the help of a linearized estimate for the nonlinear error terms. %This convergence estimate is the first such result for a third order accurate scheme for a gradient flow. Some numerical simulation results are presented to demonstrate the efficiency of the numerical scheme and the third order convergence. The long time simulation results for $\varepsilon=0.02$ (up to $T=3 \times 10^5$) have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. In particular, the power index for the surface roughness and the mound width growth, created by the third order numerical scheme, is more accurate than those produced by certain second order energy stable schemes in the existing literature.

High order numerical schemes for transport equations on bounded domains

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations

<abstract><p>In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.</p></abstract>

Numerical methods for stochastic differential equations based on Gaussian mixture

We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It\^o-Taylor expansion and iterated It\^o integrals, the proposed scheme approximates the probability measure $\mu(X^{n+1}|X^n=x_n)$ by a mixture of Gaussians. The solution at next time step $X^{n+1}$ is then drawn from the Gaussian mixture with complexity linear in the dimension $d$. This provides a new general strategy to construct efficient high weak order numerical schemes for SDEs.

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In additional, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one- and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

Innovations in Eyring–Powell radiative nanofluid flow due to nonlinear stretching sheet with convective heat and mass conditions: Numerical study

ABSTRACT The augmentation of heat and mass transfer in an industrial process due to convective boundary conditions plays an important role which cannot be ignored and the study lead to saving energy, reducing process time and increasing thermal rating. The impact of nonlinear thermal radiation on Eyring-Powell nanofluid flow over a nonlinear elongating sheet with Joule and viscous dissipation is investigated. The well-posed boundary layer problem is reduced into a highly nonlinear coupled ordinary differential system by adopting similarity transformations. The resultant equations subject to convective temperature and concentration conditions are then solved by employing a Runge-Kutta-Fehlberg 45 order numerical scheme combined with shooting technique. The flow characteristics affected by the pertinent parameters are calculated and presented through the graphs. It is witnessed that the increase in mass Biot number and Lewis number aggregates to rise in the Sherwood number. Also, the rate of heat transfer coefficient rises with growing values of thermal Biot number and radiation parameter. These investigations are of great importance in food processing, crystal growth, fibre and wire coating, production of glass fibres and in many possible areas.

Darcy–Forchheimer flow of a magneto-radiated couple stress fluid over an inclined exponentially stretching surface with Ohmic dissipation

PurposeIn this communication, a theoretical simulation is aimed to characterize the Darcy–Forchheimer flow of a magneto-couple stress fluid over an inclined exponentially stretching sheet. Stokes’ couple stress model is deployed to simulate non-Newtonian microstructural characteristics. Two different kinds of thermal boundary conditions, namely, the prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux, are considered in the heat transfer analysis. Joule heating (Ohmic dissipation), viscous dissipation and heat source/sink impacts are also included in the energy equation because these phenomena arise frequently in magnetic materials processing.Design/methodology/approachThe governing partial differential equations are transformed into nonlinear ordinary differential equations (ODEs) by adopting suitable similar transformations. The resulting system of nonlinear ODEs is tackled numerically by using the Runge–Kutta fourth (RK4)-order numerical integration scheme based on the shooting technique. The impacts of sundry parameters on stream function, velocity and temperature profiles are viewed with the help of graphical illustrations. For engineering interests, the physical implication of the said parameters on skin friction coefficient, Nussult number and surface temperature are discussed numerically through tables.FindingsAs a key outcome, it is noted that the augmented Chandrasekhar number, porosity parameter and Forchhemeir parameter diminish the stream function as well as the velocity profile. The behavior of the Darcian drag force is similar to the magnetic field on fluid flow. Temperature profiles are generally upsurged with the greater magnetic field, couple stress parameter and porosity parameter, and are consistently higher for the PEST case.Practical implicationsThe findings obtained from this analysis can be applied in magnetic material processing, metallurgy, casting, filtration of liquid metals, gas-cleaning filtration, cooling of metallic sheets, petroleum industries, geothermal operations, boundary layer resistors in aerodynamics, etc.Originality/valueFrom the literature review, it has been found that the Darcy–Forchheimer flow of a magneto-couple stress fluid over an inclined exponentially stretching surface with heat flux conditions is still scarce. The numerical data of the present results are validated with the already existing studies under limited cases and inferred to have good concord.

IRK-WSGD methods for space fractional diffusion equations

In this paper, we develop high order numerical schemes for the solution of the initial-boundary value problem of one-dimensional and two-dimensional space fractional diffusion equations of orders belonging to the interval (1,2). Firstly, certain weighted and shifted Grünwald difference (WSGD) operator is used to approximate space Riemann-Liouville fractional derivatives, resulting in a linear system of ordinary differential equations (ODEs). Then an implicit Runge-Kutta (IRK) method is applied to discretize the resulted ODEs. Thus, we get an IRK-WSGD method for the fractional diffusion equation. We prove that under certain hypotheses, the proposed IRK-WSGD schemes are stable and have temporally fourth order accuracy and spatially second/third order accuracy. Preconditioning for discretization linear systems is discussed. Numerical experiments are presented to illustrate the accuracy and efficiency of the method.

A high order numerical scheme for solving a class of non‐homogeneous time‐fractional reaction diffusion equation

AbstractThis paper is concerned with the development of a high order numerical technique for solving time‐fractional reaction–diffusion equation. The fractional derivative in the governing equation is described in the Caputo sense and a collocation method based on quintic B‐spline basis function is used to discretize the space variable. The stability and convergence analysis of the method are investigated, and it is shown that the proposed method converges to the exact solution of the problem with order of convergence O(Δt2 − α + Δx4), where α is the order of fractional derivative (0 < α < 1). Three test problems are considered to illustrate the efficiency and accuracy of present numerical scheme and to verify the theoretical result. It is shown that the order of fractional derivative has a profound effect on the solution of time‐fractional reaction–diffusion equation.

Matrix Form of Deriving High Order Schemes for the First Derivative

For many problems in Physics and Computational Fluid Dynamics (CFD), providing an accurate approximation of derivatives is a challenging task. This paper presents a class of high order numerical schemes for approximating the first derivative. These approximations are derived based on solving a special system of equations with some unknown coefficients. The construction method provides numerous types of schemes with different orders of accuracy. The accuracy of each scheme is analyzed by using Fourier analysis, which illustrates the dispersion and dissipation of the scheme. The polynomial technique is used to verify the order of accuracy of the proposed schemes by obtaining the error terms. Dispersion and dissipation errors are calculated and compared to show the features of high order schemes. Furthermore, there is a plan to study the stability and accuracy properties of the present schemes and apply them to standard systems of time dependent partial differential equations in CFD.

Derivation and Implementation of a Linear Multistep Numerical Scheme of Order 12

A great many physical occurrences give rise to problems that often result in differential equations. When we solve a differential equation, we are in effect solving the physical problem it represents. Traditionally, solutions to differential equations were derived using analytical or exact methods. These solutions are often useful as they provide excellent insight into the behavior of some systems. However, certain differential equations are very difficult to solve by any means other than an approximate solution by the application of numerical methods. These methods can be classified into two thus: One-step and multistep methods. However, in this work, our focus is on a class of multistep methods known as the Linear Multi-step Methods. We thus use Taylor series expansion to derive a linear multistep method of order 12. The method is tested for consistency and zero-stability in order to establish its convergence. Also, provided are some examples of problems solved using the new scheme, and the results compared with exact solution.

An enhanced weighted essentially non-oscillatory high order scheme for explosion modelling

Physical explosion causes large damages in the process industry and quite often escalates to chemical explosions. The shock waves generated by such events are challenging to model and they must be numerically captured without spurious oscillations in order to make an accurate estimative of the accidental effects. In this context, this paper investigates how a new high order numerical scheme models the physical explosion. We have considered a confined explosion in a spherical vessel and blast load throughout pipelines as the framework to investigate the performance of the numerical scheme. The developed numerical approach considers the effect of less smooth substencils when there is a discontinuity inside the stencil based on the local Mach number what avoids oscillations and instability. The numerical solution of the fundamental equations is coupled with the Modified Colebrook-White formulation in order to consider the blast load through the pipeline. Shock waves from experimental data and analytical model are used to validate the proposed model. The research provides an efficient method for prediction of blast loads from spherical vessels rupture to an open atmosphere and in pipelines.

A higher order numerical scheme for generalized fractional diffusion equations

In this article, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and this is the best result to date. Finally, we present four examples to confirm the theoretical results.

Exactly thermalized quantum dynamics of the spin-boson model coupled to a dissipative environment

We present an application of the Extended Stochastic Liouville-von Neumann equations (ESLN) method introduced earlier [PRB 95, 125124 (2017); PRB 97, 224310 (2018)] which describes the dynamics of an exactly thermalised open quantum system reduced density matrix coupled to a non-Markovian harmonic environment. Critically, the combined system of the open system fully coupled to its environment is thermalised at finite temperature using an imaginary time evolution procedure before the application of real time evolution. This initialises the combined system in the correct canonical equilibrium state rather than being initially decoupled. We apply our theory to the spin-boson Hamiltonian and develop a number of competing ESLN variants designed to reduce the numerical divergence of the trace of the open system density matrix. We find that a careful choice of the driving noises is essential for improving numerical stability. We also investigate the effect of applying higher order numerical schemes for solving stochastic differential equations, such as the Stratonovich-Heun scheme, and conclude that stochastic sampling dominates convergence with the improvement associated with the numerical scheme being less important for short times but required for late times. To verify the method and its numerical implementation, we consider evolution under a fixed Hamiltonian and show that the system either remains in, or approaches, the correct canonical equilibrium state at long times. Additionally, evolution of the open system under non-equilibrium Landau-Zener (LZ) driving is considered and the asymptotic convergence to the LZ limit was observed for vanishing system-environment coupling and temperature. When coupling and temperature are non-zero, initially thermalising the combined system at a finite time in the past was found to be a better approximation of the true LZ initial state than a pure state.

Weak Galerkin finite element with curved edges

This paper studies a weak Galerkin finite element method on a domain with curved edges. We consider the two dimensional Poisson’s equation of a fixed curved boundary. Usually, the straight polygon meshes deliver a sub-optimal rate of convergence for the curved domain as approximation order bounded by O(h2) even for high order numerical schemes. In this paper, we shall develop a high order (optimal rate of convergence) WGFEM based on the curved mesh. The implementation details are studied for efficient numerical integration tools. Numerical experiments are provided to show the proposed curved WGFEM produces an optimal rate for errors measured in L2-norm and energy-norm of convergence as O(hk+1) and O(hk) for curved two-dimensional domain.

An improved Roe solver for high order reconstruction schemes

Roe solver is one of the most popular upwind schemes in computational fluid dynamics which is first-order accurate in space. However, the amount of numerical dissipation gets larger than necessary for high order numerical schemes and then becomes the dominant term of the numerical error. This paper proposes a modified Roe solver that can use high order schemes with high stability based on splitting of the upwind term. The novel algorithm is proposed to compute the dissipation term of Roe flux function using low and high reconstruction schemes for computing acoustic and entropy waves information, respectively. It changes the original Roe solver dissipation term to improve the level of accuracy. The third-order WENO-Z and MUSCL schemes are adopted as high reconstruction schemes. Finite difference approach is used to simulate Euler equations with the original and modified Roe solvers. The results of four shock tube problems of Sod, Lax, Mach 3, and supersonic test cases are compared with the exact solution of the Riemann problem. It is found that the proposed Roe solver increases the accuracy and robustness of the original Roe solver, especially around the expansion waves, contact discontinuities, and shock waves.

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

PurposeThis paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.Design/methodology/approachIn this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.FindingsThe paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.Research limitations/implicationsThere are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.Practical implicationsPhysical problems with singular and non-singular coefficients are directly solved by this method.Originality/valueThe paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

A uniformly convergent finite difference scheme for Robin type singularly perturbed parabolic convection diffusion problem

In this paper, a second order numerical scheme for solving a singularly perturbed convection diffusion problem with Robin boundary conditions is proposed. The numerical scheme is a combination of the fitted operator finite difference and the backward Euler finite difference methods. These are designed in order to solve respectively the spatial derivatives and the time derivative. Using some properties of the discrete problem the methods are analysed for convergence. Richardson extrapolation technique is used to improve the accuracy and also accelerate the convergence of the method. Numerical simulations are carried out to confirm the theoretical findings in the analysis before and after extrapolation.

A higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers

In this article, a parameter-uniform numerical method is presented to solve one-dimensional singularly perturbed parabolic convection-diffusion multiple turning point problems exhibiting two exponential boundary layers. We study the asymptotic behaviour of the solution and its partial derivatives. The problem is discretized using the implicit Euler method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh. The scheme is shown to be ε-uniformly convergent of order one in time direction and order two in spatial direction upto a logarithmic factor. Numerical experiments are conducted to validate the theoretical results. Comparison is done with the upwind scheme on a uniform mesh as well as on the standard Shishkin mesh to demonstrate the higher order accuracy of the proposed scheme on a generalized Shishkin mesh.

High Order Numerical Simulations for the Binary Fluid--Surfactant System Using the Discontinuous Galerkin and Spectral Deferred Correction Methods

In this paper, we propose a high order numerical scheme to simulate the binary fluid--surfactant system by combining the semi-implicit spectral deferred correction (SDC) method and the energy stabl...