Classical Finite Difference Scheme Research Articles

On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $\alpha$ reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy model, while there exists a $-\alpha/3$ power law coarsening stage.

A parameter uniform essentially first-order convergent numerical method for a parabolic system of singularly perturbed differential equations of reaction–diffusion type with initial and Robin boundary conditions

In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction–diffusion type with initial and Robin boundary conditions is considered. The components of the solution [Formula: see text] of this system are smooth, whereas the components of [Formula: see text] exhibit parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.

A non-conventional hybrid numerical approach with multi-dimensional random sampling for cocaine abuse in Spain

This paper introduces a non-conventional approach with multi-dimensional random sampling to solve a cocaine abuse model with statistical probability. The mean Latin hypercube finite difference (MLHFD) method is proposed for the first time via hybrid integration of the classical numerical finite difference (FD) formula with Latin hypercube sampling (LHS) technique to create a random distribution for the model parameters which are dependent on time [Formula: see text]. The LHS technique gives advantage to MLHFD method to produce fast variation of the parameters’ values via number of multidimensional simulations (100, 1000 and 5000). The generated Latin hypercube sample which is random or non-deterministic in nature is further integrated with the FD method to complete one cycle of LHS-FD simulation iteration. This process is repeated until [Formula: see text] final iterations of LHS-FD are obtained. The means of these [Formula: see text] final solutions (MLHFD solutions) are tabulated, graphed and analyzed. The numerical simulation results of MLHFD for the SEIR model are presented side-by-side with deterministic solutions obtained from the classical FD scheme and homotopy analysis method with Pade approximation (HAM-Pade). The present MLHFD results are also compared with the previous non-deterministic statistical estimations from 1995 to 2015. Good agreement between the two is perceived with small errors. MLHFD method can be used to predict future behavior, range and prediction interval for the epidemic model solutions. The expected profiles of the cocaine abuse subpopulations are projected until the year 2045. Both the statistical estimations and the deterministic results of FD and HAM-Pade are found to be within the MLHFD prediction intervals for all the years and for all the subpopulations considered.

Numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem

A singularly perturbed problem involving two singular perturbation parameters is discretized using the classical upwinded finite difference scheme on an appropriate piecewise-uniform Shishkin mesh. Scaled discrete derivatives (with scaling only used within the layers) are shown to be parameter uniformly convergent to the scaled first derivatives of the continuous solution.

Chebyshev multidomain pseudospectral method to solve cardiac wave equations with rotational anisotropy

The implementation of numerical methods to solve and study equations for cardiac wave propagation in realistic geometries is very costly, in terms of computational resources. The aim of this work is to show the improvement that can be obtained with Chebyshev polynomials-based methods over the classical finite difference schemes to obtain numerical solutions of cardiac models. To this end, we present a Chebyshev multidomain (CMD) Pseudospectral method to solve a simple two variable cardiac models on three-dimensional anisotropic media and we show the usefulness of the method over the traditional finite differences scheme widely used in the literature.

Fractional Step Method for Singularly Perturbed 2D Delay Parabolic Convection Diffusion Problems on Shishkin Mesh

In this article, we are interested to approximate the solution of a singularly perturbed 2D delay parabolic convection–diffusion initial-boundary-value problem. To discretize the continuous problem in the temporal direction, we use a fractional step method which results a set of two 1D problems. Next, we apply classical finite difference scheme on a special mesh to discretize those 1D problems in the spatial directions. Fractional step method for the time variable permits the computational cost reduction and the special mesh is used to capture the boundary layers. We derive the truncation errors for the scheme to obtain the error estimates, which shows that the scheme is uniformly convergent of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are presented to support the theoretical results.

Semi-implicit staggered discontinuous Galerkin schemes for axially symmetric viscous compressible flows in elastic tubes

We propose a novel family of staggered semi-implicit discontinuous Galerkin (DG) finite element schemes for the simulation of axially symmetric, weakly compressible and laminar viscous flows in elastic pipes. The equation of state (EOS) of the fluid is assumed to be barotropic and two different mathematical models derived from the compressible Navier-Stokes equations are considered in this paper.The first model describes cross-sectionally averaged 1D flows, including steady and frequency-dependent wall friction effects. The novelty of our numerical method compared to standard DG schemes consists in the use of a staggered mesh, where the pressure is defined over a primary grid and the velocity field is defined on edge-based staggered dual control volumes. This approach is well known from classical semi-implicit finite difference schemes for the incompressible Navier–Stokes equations, but it is still quite unusual for high order DG schemes. The continuity equation is integrated over the control volumes that belong to the main grid, while the momentum equation is integrated over the elements of the edge-based staggered dual grid. The nonlinear convective terms are discretized explicitly, while the pressure gradient and the mass flux are discretized implicitly. Up to second order of accuracy in time can be achieved with the so-called θ-method. Inserting the discrete momentum equation in the discrete mass conservation equation leads to a mildly nonlinear algebraic system for the degrees of freedom of the pressure. Such mildly nonlinear systems can be very efficiently solved using the Newton algorithm of Brugnano and Casulli. We observe that the linear part of the mildly nonlinear system is symmetric and positive definite.The second model is derived from the compressible Navier-Stokes equations in cylindrical coordinates. Assuming hydrostatic flow with constant pressure inside each cross section as well as axial symmetry, only the terms in the axial and the radial direction need to be considered. Therefore, we call the second model the 2Dxr model. Also in this case we use a staggered mesh for pressure and velocity and thus the same philosophy as for the 1D model can be applied to obtain the discrete pressure system. For the 2Dxr model a staggered DG scheme is also applied for the computation of the viscous stress tensor in the discrete momentum equation. However, in radial direction the resulting linear system for the friction terms is not symmetric and is thus solved using the Thomas algorithm for block three-diagonal systems.The use of a semi-implicit DG scheme leads to a very mild CFL condition based only on the fluid velocity and not on the sound speed, which makes the method very efficient, in particular in the limit cases when the speed of sound of the fluid tends to infinity (incompressible fluid) and in the rigid case where the wall strain of the pipe tends to zero. In addition, at every Newton step a symmetric positive definite and well conditioned block three-diagonal linear system is solved for the pressure, using a matrix-free conjugate gradient method. Moreover, when the polynomial degree of the basis and test functions is equal to zero the schemes reduce to classical semi-implicit finite volume methods.While in the 2Dxr model the viscous effects in radial direction are directly obtained from first principles via the Navier-Stokes equations, the 1D model requires an additional closure relation for the wall friction. For both models we perform several tests in order to validate the numerical methods for steady and unsteady flows of compressible and nearly incompressible fluids in elastic and rigid tubes. We also provide numerical convergence results in order to show that the developed schemes achieve high order of accuracy in space.

Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. I. Explicit Methods

We consider specific realizations of different implicit generalized Runge–Kutta methods as applied to the numerical integration with respect to time of initial-boundary-value problems for the second-order parabolic equations and investigate their spectral stability. It is shown that all implicit generalized Runge–Kutta methods are unconditionally spectrally stable but some of them have the conditional property of monotonicity of the numerical solution with respect to time. The functions of spectral stability of the implicit generalized Runge–Kutta methods are rational. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different implicit generalized Runge–Kutta methods. It is shown that, in this case, the application of the one-stage Radau methods with subsequent discretization of the problem with respect to the space variable leads to the classical forward finite difference scheme (Laasonen scheme), whereas the use of the one-stage Gauss–Legendre method leads to a six-point symmetric scheme (Crank–Nicolson scheme). It is shown that diagonally implicit generalized Norsett and Burrage methods are realized in almost the same way as the one-stage Radau and Gauss–Legendre methods but their accuracy in the time step is 10–1000 times higher. On the basis of comparison of the numerical and analytic solutions, we conclude that, in order to get practically suitable numerical solutions without any restrictions on the time step, it is reasonable to use one- and three-stage generalized Radau methods or two- and four-stage Lobatto IIIC methods. All other explicit and implicit generalized Runge–Kutta methods require certain restrictions imposed on the time step.

Higher-order convergence with fractional-step method for singularly perturbed 2D parabolic convection–diffusion problems on Shishkin mesh

In this article, we propose a second-order uniformly convergent numerical method for a singularly perturbed 2D parabolic convection–diffusion initial–boundary-value problem. First, we use a fractional-step method to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction, which gives a set of two 1D problems. Then, we use the classical finite difference scheme to discretize those 1D problems on a special mesh, which results almost first-order convergence, i.e., O(N−1+βlnN+Δt). To enhance the order of convergence to O(N−2+βln2N+Δt2), we use the Richardson extrapolation technique. In support of the theoretical results, numerical experiments are performed by employing the proposed technique.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Scalar Convection Diffusion Equations

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak...

Numerical approximation of 2D time dependent singularly perturbed convection–diffusion problems with attractive or repulsive turning points

In this work, we are interested in approximating the solution of 2D parabolic singularly perturbed problems of convection–diffusion type. The convective term of the differential equation, associated to the initial and boundary value problem, is such that each one of its components has an interior simple turning point, which can be of attractive or repulsive type. We describe a numerical method to discretize the continuous problem, which combines the fractional implicit Euler method, defined on a uniform mesh, to discretize in time, and the classical upwind finite difference scheme, defined on a nonuniform mesh of Shishkin type, to discretize in space. The fully discrete algorithm has a low computational cost. From a numerical point of view, we see that the method is efficient and uniformly convergent with respect to the diffusion parameter in both cases when the source term is a continuous function or it has a first kind discontinuity at the turning points. Some numerical results for different test problems are showed; from them, we deduce the good properties of the numerical method.

Algorithm 986

A collection of Matlab routines that compute derivative approximations of arbitrary functions using high-order compact finite difference schemes is presented. Tenth-order accurate compact finite difference schemes for first and second derivative approximations and sixth-order accurate compact finite difference schemes for third and fourth derivative approximations are discussed for the functions with periodic boundary conditions. Fourier analysis of compact finite difference schemes is explained, and it is observed that compact finite difference schemes have better resolution characteristics when compared to classical finite difference schemes. Compact finite difference schemes for the functions with Dirichlet and Neumann boundary conditions are also discussed. Moreover, compact finite difference schemes for partial derivative approximations of functions in two variables are also given. For each case a Matlab routine is provided to compute the differentiation matrix and results are validated using the test functions.

Uniform discrete Sobolev estimates of solutions to finite difference schemes for singular limits of nonlinear PDEs

Uniform discrete Sobolev space estimates are proven for a class of finite-difference schemes for singularly-perturbed hyperbolic-parabolic systems. The estimates obtained improve previous results even when the PDEs do not involve singular perturbations. These estimates are used in a companion paper to prove the convergence of solutions as the discretization parameter and/or the singular perturbation parameter tends to zero.

Comparison between numerical methods applied to the damped wave equation

For the numerical solution of Dirichlet-Neumann problems related to 1D damped wave propagation, from a numerical point of view, we compare the so-called energetic approach, considered here separately for boundary and finite element methods with classical finite difference schemes, both explicit and implicit. The analysis reveals the superiority of energetic approximations with respect to unconditional stability and accuracy with respect to any choice of discretization parameters.

Adjoint design sensitivity analysis and optimization of nonlinear structures using geometrical mapping approach

A new adjoint shape design sensitivity formulation for nonlinear structures subject to contact forces is developed. The method is based on a geometrical mapping approach where shape variation is regarded as a mapping characterized by the shape variation velocity field. An adjoint variable method is developed for performing sensitivity analysis of the average shear strain in the tire belt area, which profile is properly parameterized in function of arcs, allowing explicit design velocities calculated. It can be seen that the present method is much faster than the direct differentiation method and more accurate than the classical finite difference scheme.

An Adaptive Grid Method for Singularly Perturbed Time-Dependent Convection-Diffusion Problems

AbstractIn this paper, we study the numerical solution of singularly perturbed time-dependent convection-diffusion problems. To solve these problems, the backward Euler method is first applied to discretize the time derivative on a uniform mesh, and the classical upwind finite difference scheme is used to approximate the spatial derivative on an arbitrary nonuniform grid. Then, in order to obtain an adaptive grid for all temporal levels, we construct a positive monitor function, which is similar to the arc-length monitor function. Furthermore, the ε-uniform convergence of the fully discrete scheme is derived for the numerical solution. Finally, some numerical results are given to support our theoretical results.

A fractional step method for 2D parabolic convection-diffusion singularly perturbed problems: uniform convergence and order reduction

In this work, we are concerned with the efficient resolution of two dimensional parabolic singularly perturbed problems of convection-diffusion type. The numerical method combines the fractional implicit Euler method to discretize in time on a uniform mesh and the classical upwind finite difference scheme, defined on a Shishkin mesh, to discretize in space. We consider general time-dependent Dirichlet boundary conditions, and we show that classical evaluations of the boundary conditions cause an order reduction in the consistency of the time integrator. An appropriate correction for the evaluations of the boundary data permits to remove such order reduction. Using this correction, we prove that the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. Some numerical experiments, which corroborate in practice the robustness and the efficiency of the proposed numerical algorithm, are shown; from them, we bring to light the influence in practice of the two options for the boundary data considered here, which is in agreement with the theoretical results.

A parameter uniform almost first order convergent numerical method for non-linear system of singularly perturbed differential equations

In this paper an initial value problem for a non-linear system of two singularly perturbed first order differential equations is considered on the interval (0,1].The components of the solution of this system exhibit initial layers at 0. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be almost first order convergent in the maximum norm uniformly in the perturbation parameters.

English

This paper presents a physical model which describes the layer growth kinetics and the nitrogen concentration profiles during gaseous or plasma nitriding of pure iron. The model is related to a one dimensional moving boundary value problem where the initial concentration profiles are assumed to be linear and a new boundary condition at the diffusion zone is proposed. The model is solved by using a classical finite difference scheme (FDM-CS) and the Heat Balance Integral Method (HBIM). Due to the proposed boundary condition at the diffusion zone, the numerical solutions can be validated through a simple physical analysis of mass transport theory in the asymptotic time limit. The results obtained are compared, and observed to be in good agreement with available experimental data and other approximate solutions reported in the literature. Key words: Advanced characterization, coatings, plasma nitriding, finite difference, heat balance.

Parameter uniform numerical method for fourth order singularly perturbed turning point problems exhibiting boundary layers

Abstract In this paper, a numerical method based on Shishkin mesh for a singularly perturbed fourth order differential equation with a turning point exhibiting boundary layers is presented. In this method the problem is transformed into a weakly coupled system of two second order equations, one without the parameter e and the other with the parameter e multiplying the highest derivative with suitable boundary conditions. We apply classical finite difference scheme on an appropriate piecewise uniform (Shishkin) mesh. Parameter uniform error bounds for the numerical solution are established. Numerical results are provided to illustrate the theoretical results.