Class Of Difference Schemes Research Articles

Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases

AbstractThis is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.

A New Class of Difference Schemes With Intrinsic Parallelism for the KdV-Burgers Equation

A New Class of Difference Schemes With Intrinsic Parallelism for the KdV-Burgers Equation

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

A graded mesh refinement approach for boundary layer originated singularly perturbed time‐delayed parabolic convection diffusion problems

In this work, we consider a graded mesh refinement algorithm for solving time‐delayed parabolic partial differential equations with a small diffusion parameter. The presence of this parameter leads to boundary layer phenomena. These problems are also known as singularly perturbed problems. For these problems, it is well‐known that one cannot achieve a convergent solution to maintain the boundary layer dynamics, on a fixed number of uniform meshes irrespective of the arbitrary magnitude of perturbation parameter. Here, we consider an adaptive graded mesh generation algorithm, which is based on an entropy function in conjunction with the classical difference schemes, to resolve the layer behavior. The advantage of the present algorithm is that it does not require to have any information about the location of the layer. Several examples are presented to show the high performance of the proposed algorithm.

Об инвариантных конечно-разностных схемах для уравнений одномерных течений политропного газа для задач с пространственными симметриями

One-dimensional polytropic gas dynamics equations for plane, radially symmetric, and spherically symmetric flows are considered. Invariant properties of equations are discussed, local conservation laws are derived. Additional conservation laws are written, which take place only in case of special values of adiabatic exponent. Classical difference scheme of Samarsky-Popov for gas dynamics has all difference analogs of conservation laws, except for additional ones. In difference schemes additional conservative laws take place in case of special state equation approximation. Scheme of Samarsky-Popov with special state equation was initially suggested by V.A. Korobitsyn. He described it as ‘thermodynamically consistend’ In current paper group properties, and conservation laws of thermodynamically consistent schemes are discussed, and numerical implementation for plane, cylinder, and spherical flows is perfomed.

Time-fractional diffusion equation-based image denoising model

In the last two decades, due to the use of total variation (TV) denoising, variation models for image reconstruction have achieved great success. Compare to some known variation models which have just adopted the space-fractional diffusion equation, the time-fractional diffusion equation has another adjustable time-fractional derivative order to control the diffusion process. Therefore, in this paper, a new denoising model based on time-fractional diffusion equation is proposed, where the discretization in space is the application of classical difference scheme while the discretization in time is the approximation of the Caputo differential. Furthermore, both the explicit and implicit numerical scheme named TFEIS are built and its stability and convergence are rigorously discussed. We prove the full-discretization is stable under some conditions and the numerical solution converges to the exact one with order $$O(\tau^{2 - \alpha } + h^{2} )$$ , where τ and $$h$$ are the time and space step size, respectively. Finally, various evaluation indexes such as histogram recovery degree, signal-to-noise ratio, and edge texture retention index are applied to compare new and original models comprehensively in lots of images. Computer simulation results show our models

Compact Finite Difference Schemes for Approximating Differential Relations

Differential relations include both differential operators and solvers for boundary value problems. The formulas of compact finite difference approximations for first- and second-order differential relations of the form $${{P}_{1}}[u] = {{P}_{2}}[f]$$ are obtained. An approximation on three-point stencils is used. Its implementation, as in the case of classical difference schemes, requires the inversion of a tridiagonal matrix. However, compact schemes provide significantly higher accuracy and the fourth order of approximation instead of the second.

ROBUST DIFFERENCE SCHEME FOR THE CAUCHY PROBLEM FOR A SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATION

Grid approximation of the Cauchy problem on the interval D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem under consideration, for small values of the parameter ε, a boundary layer of width O(ε) appears on which the solution varies by a finite value. It is shown that, for such a Cauchy problem, the solution of the standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm; convergence occurs only under the condition h ε, where h = d N −1 , N is the number of grid intervals, h is the grid step-size. Taking into account the behavior of the singular component in the solution, a special piecewise-uniform grid is constructed that condenses in a neighborhood of the boundary layer. It is established that the standard difference scheme on such a special grid converges ε-uniformly in the maximum norm at the rate O(N −1 lnN). Such a scheme is called a robust one. For a model Cauchy problem for a singularly perturbed ordinary differential equation, standard difference schemes on a uniform grid (a classical difference scheme) and on a piecewise-uniform grid (a special difference scheme) are constructed and investigated. The results of numerical experiments are given, which are consistent with theoretical results.

On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations

For a broad class of initial-boundary value problems for quasilinear parabolic equations with nonlinear source and their approximations, we show that if the initial energy is negative, then the solution always blows up in finite time. This is especially important for finding sufficiently simple and easy-to-verify conditions guaranteeing the presence of physical effects such as heat localization in peaking modes or thermal explosions and for deriving two-sided estimates of the solution lifespan. We construct the corresponding new classes of difference schemes for which grid analogs of integral conservation laws hold. We show that, to obtain efficient two-sided estimates for the blow-up time of the solution of the differential problem, in practice, one should use difference schemes with explicit as well as implicit approximation to the source.

Mixed spline function method for reaction–subdiffusion equations

In this paper, we develop two classes of finite difference schemes for the reaction–subdiffusion equations by using a mixed spline function in space direction, forward and backward difference in time direction, respectively. It has been shown that some of the previous known difference schemes can be derived from our schemes if we suitably choose the spline parameters. By Fourier method, we prove that one class of difference scheme is unconditionally stable and convergent, the other is conditionally stable and convergent. Finally, some numerical results are provided to demonstrate the effectiveness of the proposed difference schemes.

Operator-difference schemes for a class of systems of evolution equations

For a special system of evolution equations of first order, discrete time approximations for the approximate solution of the Cauchy problem are considered. Such problems arise after the spatial approximation in the Schrodinger equation and the subsequent separation of the imaginary and real parts and in nonstationary problems of acoustics and electrodynamics. Unconditionally stable two time level operator-difference weighted schemes are constructed. The second class of difference schemes is based on the formal passage to explicit operator-difference schemes for evolution equations of second order when explicit-implicit approximation is used for isolated equations of the system. The regularization of such schemes in order to obtain unconditionally stable operator difference schemes is discussed. Splitting schemes involving the solution of simplest problems at each time step are constructed.

GRID APPROXIMATION OF SINGULARLY PERTURBED PARABOLIC EQUATIONS WITH MOVING BOUNDARY LAYERS

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary layer appears in a neighbourhood of the left lateral boundary SL 1. It turns out that, in the class of difference schemes on rectangular grids condensing in a neighbourhood of SL 1 with respect to x and t, there do not exist schemes that converge even under the condition P 0 −1 Â ϵ1/2, where P 0 is the total number of nodes in the meshes used, that is, P 0 Â N N 0, where the values N and N 0 define the numbers of mesh points in x and t. On such meshes, convergence under the condition N −1 + N 0 −1 ≤ ϵ1/4 cannot be achieved. Examination of widths similar to Kolmogorov's widths allows us to establish necessary and sufficient conditions for the ϵ‐uniform convergence of approximations to the solution of the boundary value problem. Using these conditions, a scheme is constructed on a mesh being piece‐wise uniform in a coordinate system adapted to the moving boundary. This scheme converges ϵ‐uniformly at the rate O(N −1 ln N + N0 −1).

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ɛ (this is the coefficient of the higher order derivatives of the equation, ɛ ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S1L. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ɛ-uniformly at a rate of O(N−1lnN + N0), where N and N0 define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N−1 + N0−1 ≪ ɛ. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S1L with respect to x and t, the convergence under the condition N−1 + N0−1 ≤ ɛ1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ɛ-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ɛ-uniformly at a rate of O(N−1lnN + N0).

Construction of Space-Time High Resolution Second Order Accurate Three Dimensional MP Difference Schemes

In this paper, a new Riemann-solver-free class of difference schemes are constructed to scalar nonlinear hyperbolic conservation laws in the three dimension (3D). We proved that these schemes had second order accuracy in space and time, and satisfied maximum principles (marked as MPs) under an appropriate CFL condition. This results in a second-order accuracy, MP schemes a natural extension of the one (two)-dimensional second-order. In addition, these schemes can still be extended to the vector system of conservation law. We yet prove that these schemes satisfied the scalar and vector maximum principle, and in the more general context of systems.

A high resolution total variation diminishing scheme for hyperbolic conservation law and related problems

A high resolution, explicit second order accurate, total variation diminishing (TVD) difference scheme using flux limiter approach for scalar hyperbolic conservation law and closely related convectively dominated diffusion problem is presented. Bounds and TVD region is given for these limiters such that the resulting scheme is TVD. A class of such limiters is given which gives second order accurate TVD scheme. Numerical results for one-dimensional scalar conservation law and convectively dominated diffusion problem are presented. Comparison of numerical results with some of the classical difference schemes are given.

A class of difference schemes with flexible local approximation

Solutions of many physical problems have salient local features that are qualitatively known a priori (for example, singularities at point sources, edge and corners; boundary layers; derivative jumps at material interfaces; strong dipole field components near polarized spherical particles; cusps of electronic wavefunctions at the nuclei; electrostatic double layers around colloidal particles, etc.) The known methods capable of providing flexible local approximation of such features include the generalized finite element – partition of unity method, special variational-difference schemes in broken Sobolev spaces, and a few other specialized techniques. In the proposed new class of Flexible Local Approximation MEthods (FLAME), a desirable set of local approximating functions (such as cylindrical or spherical harmonics, plane waves, harmonic polynomials, etc.) defines a finite difference scheme on a chosen grid stencil. One motivation is to minimize the notorious ‘staircase’ effect at curved and slanted interface boundaries. However, the new approach has much broader applications. As illustrative examples, the paper presents arbitrarily high order 3-point schemes for the 1D Schrödinger equation and a 1D singular equation, schemes for electrostatic interactions of colloidal particles, electromagnetic wave propagation and scattering, plasmon resonances. Moreover, many classical finite difference schemes, including the Collatz “Mehrstellen” schemes, are direct particular cases of FLAME.

Limitations of adaptive mesh refinement techniques for singularly perturbed problems with a moving interior layer

In a composed domain on an axis R with the moving interface boundary between two subdomains, we consider an initial value problem for a singularly perturbed parabolic reaction–diffusion equation in the presence of a concentrated source on the interface boundary. Monotone classical difference schemes for problems from this class converge only when ε⪢ N −1+ N 0 −1, where ε is the perturbation parameter, N and N 0 define the number of mesh points with respect to x (on segments of unit length) and t. Therefore, in the case of such problems with moving interior layers, it is necessary to develop special numerical methods whose errors depend rather weakly on the parameter ε and, in particular, are independent of ε (i.e., ε-uniformly convergent methods). In this paper we study schemes on adaptive meshes which are locally condensing in a neighbourhood of the set γ ∗ , that is, the trajectory of the moving source. It turns out, that in the class of difference schemes consisting of a standard finite difference operator on rectangular meshes which are (a priori or a posteriori) locally condensing in x and t, there are no schemes that converge ε-uniformly, and in particular, even under the condition ε≈ N −2+ N 0 −2, if the total number of the mesh points between the cross-sections x 0 and x 0+1 for any x 0∈ R has order of NN 0. Thus, the adaptive mesh refinement techniques used directly do not allow us to widen essentially the convergence range of classical numerical methods. On the other hand, the use of condensing meshes but in a local coordinate system fitted to the set γ ∗ makes it possible to construct schemes which converge ε-uniformly for N, N 0→∞ ; such a scheme converges at the rate O(N −1 ln N+N 0 −1) .

On mean value solutions for the Helmholtz equation on square grids

A numerical treatment for the boundary value problem involving the Helmholtz equation Δu− λ 2 u= f is presented. The method is a five-point formula with an improved accuracy when compared with the usual finite difference method. Besides, the accuracy evaluation is provided in analytical form and the classical difference scheme is seen as a truncated series approximation to the present method. The idea comes from approximations to analytical solutions to the Dirichlet problem inside a ball, based on the Green identity. The homogeneous and the nonhomogeneous parts are evaluated in separate expressions, and the precision error yielded is of order O( h 2). Some numerical examples and comparisons are presented.

The DAE-aspect of the charge cycle in a combustion engine

The gas exchange in the cylinders is essential for the overall performance of an internal combustion engine. A CAE approach leads to technically improved and less polluting engines. Its base is a network formulation of the engine. The mathematical model describes the gas flow in the various parts of the engine. A system of ordinary differential equations in time represents conservation of mass and energy in the chambers. Considering the gas flow in the pipes yields an hyperbolic system of partial differential equaitons, the unsteady one-dimensional Euler equations. In contrast to classical difference schemes the numerical treatment with an ENO scheme prevents spurious numerical oscillations. The coupling of the pipes with the cylinders and other elements of the network leads to differential-algebraic equations (DAEs), which are solved by a predictor-corrector method. The implementation in an industrial simulation package shows that even large-scale problems can be solved with a very good congruence of measured and computed data.

A class of stable difference schemes for linear elliptic PDEs and their asynchronous parallel computation

This paper improves and generalizes the two difference schemes presented in paper [1] and gives a new difference scheme for second order linear elliptic partial differential equations, its difference matrix is a M-matrix and because of the stability of the M-matrix, it is convergent by the asynchronous iterative method on multiprocessors. Then this paper gives a class of difference schemes for linear elliptic PDEs so that their difference matrixes are all M-matrixes and their asynchronous parallel computation are convergent.