Piecewise Uniform Grids Research Articles

Optimization of calculations in modeling the breaking of plasma oscillations

The proposed method of optimization of calculations is of a combined nature. First, using a piecewise uniform grid, a difference scheme of the second order of accuracy of the McCormack type is constructed, and then — to weaken the stability condition — an implicit version of the scheme is used, which even in the nonlinear case allows an iteration-free implementation. When constructing a grid, a preliminary study is required, which is based on the analytical properties of the solution of the differential problem. Based on the calculations carried out, it can be concluded that the potential reduction in the amount of calculations is dozens of times without a noticeable loss of accuracy.

A posteriori grid method for a time-fractional Black-Scholes equation

<abstract><p>In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $ L1 $ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $ L1 $ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.</p></abstract>

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.

Application of non-local transformations for numerical integration of singularly perturbed boundary-value problems with a small parameter

Singularly perturbed boundary-value problems for second-order ODEs of the form εyxx′′=F(x,y,yx′) with ε→0 are considered. We present a new method of numerical integration of such problems, based on introducing a new non-local independent variable ξ, which is related to the original variables x andy by the equation ξx′=g(x,y,yx′,ξ). With a suitable choice of the regularizing function g, this method leads to more appropriate problems that allow the application of standard numerical methods with fixed stepsize of ξ (in the whole range of variation of the independent variable x, including both the boundary-layer region and the outer region). It is shown that methods based on piecewise-uniform grids are a particular (degenerate) case of the method of non-local transformations with a piecewise-smooth regularizing function of special form. A number of linear and non-linear test problems with a small parameter (including convective heat and mass transfer type problems) that have exact or asymptotic solutions (both monotonic and non-monotonic), expressed in elementary functions, are presented. Comparison of numerical, exact, and asymptotic solutions showed the high efficiency of the method of non-local transformations for solving singularly perturbed problems with boundary layers. In addition to non-local transformations, examples of the use of point (local) transformations for numerical integration of singularly perturbed boundary-value problems are also given.

On the parabolic spline-interpolation of functions with a boundary layer

A subject of the article is a parabolic spline-interpolation of functions having high gradient domains. Uniform grids’ inefficiency was shown. The piecewise-uniform grids, concentrated in the boundary layer were scrutinized. There were announced asymptotically exact estimates on a class of functions with an exponential boundary layer in consideration with the traditional parabolic spline interpolation. There were obtained results showing non-uniform in small parameter estimates. A probable divergence of interpolation processes was extrapolated. A modified parabolic spline with obtained uniform in small parameter interpolation error estimations was introduced. The paper contains results of numerical experiments, which strengthen theoretical estimates.

Difference scheme of highest accuracy order for a singularly perturbed reaction–diffusion equation based on the solution decomposition method

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction–diffusion equation. For this problem, a new approach is developed in order to construct difference schemes whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter e, e ∈ (0, 1] (i.e., e-uniformly) with order of accuracy significantly greater than the ultimate achievable accuracy order for the Richardson method on piecewise uniform grids. The main point of this approach is that uniform grids are used to solve grid subproblems for the regular and singular components of the discrete solution. Using the asymptotic construction technique, a basic difference scheme of the solution decomposition method is constructed that converges e-uniformly in the maximum norm at the rate O(N −2 ln2 N), where N + 1 is the number of nodes in the uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges e-uniformly in the maximum norm at the rate O(N −6 ln6 N).

Lagrange interpolation and Newton-Cotes formulas for functions with boundary layer components on piecewise-uniform grids

The problem of interpolation of a one-variable function considered as a solution to a boundary value problem for an equation with a small parameter ɛ in the highest derivative is investigated. The application of a Lagrange polynomial for such a function on a uniform grid may result in serious errors. For a Shishkin grid, ɛ-uniform error estimates of Lagrange polynomial interpolation are obtained. The Shishkin grid is modified to increase the interpolation accuracy. For Newton-Cotes formulas on such grids, ɛ-uniform error estimates are obtained. The results of numerical experiments confirm the theoretical estimates.

Extrapolation cascadic multigrid method on piecewise uniform grid

The triangular linear finite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed. Global superconvergence in discrete H 1-norm and global extrapolation in discrete L 2-norm are proved. Based on these global estimates the conjugate gradient method (CG) is effective, which is applied to extrapolation cascadic multigrid method (EXCMG). The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.

Convergence Analysis of Weighted Difference Approximations on Piecewise Uniform Grids to a Class of Singularly Perturbed Functional Differential Equations

We consider the numerical approximation of a singularly perturbed time delayed convection diffusion problem on a rectangular domain. Assuming that the coefficients of the differential equation be smooth, we construct and analyze a higher order accurate finite difference method that converges uniformly with respect to the singular perturbation parameter. The method presented is a combination of the central difference spatial discretization on a Shishkin mesh and a weighted difference time discretization on a uniform mesh. A priori explicit bounds on the solution of the problem are established. These bounds on the solution and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. It is shown that the proposed method is \(L_{2}^{h}\)-stable. The analysis done permits its extension to the case of adaptive meshes which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practical satisfies the theoretical predictions.

Iterative Newton solution method for the Richardson scheme for a semilinear singular perturbed elliptic convection–diffusion equation

Grid approximations of the Dirichlet problem are considered in a vertical strip for the semilinear elliptic convection–diffusion equation; for this problem, the nonlinear difference scheme based on the classic approximation of the problem on a piecewise-uniform grid refined in a layer converges ε-uniformly in the uniform norm with the convergence rate order not higher than one.

Conditional ɛ-uniform convergence of adaptive algorithms in the finite element method as applied to singularly perturbed problems

The boundary value problem for the ordinary differential equation of reaction-diffusion on the interval [−1, 1] is examined. The highest derivative in this equation appears with a small parameter ɛ2 (ɛ ∈ (0, 1]). As the small parameter approaches zero, boundary layers arise in the neighborhood of the interval endpoints. An algorithm for the construction of a posteriori adaptive piecewise uniform grids is proposed. In the adaptation process, the edges of the boundary layers are located more accurately and the grid on the boundary layers is repeatedly refined. To find an approximate solution, the finite element method is used. The sequence of grids constructed by the algorithm is shown to converge “conditionally ɛ-uniformly” to some limit partition for which the error estimate O(N−2ln3N) is proved. The main results are obtained under the assumption that ɛ ≪ N−1, where N is number of grid nodes; thus, conditional ɛ-uniform convergence is dealt with. The proofs use the Galerkin projector and its property to be quasi-optimal.

A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N 1 −2 ln2 N 1 + N 2 −2 ), where N 1 + 1 and N 2 + 1 are the number of grid nodes along the x 1-axis and per unit interval of the x 2-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ɛ2, where ɛ ∈ (0, 1]. When ɛ is small, a boundary and an interior layer (with the characteristic width ɛ) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ɛ, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ɛ-uniformly at a rate of O(N−2ln2N + n0−1), where N + 1 and N0 + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ɛ-uniformly at a rate of O(N−3 + N0−2) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ɛ-uniformly in x with an order greater than three.

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdo- main in which the grid solution should be further refined given the perturbation parameter e , the size of the uniform mesh in x , the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on e . The scheme converges almost e -uniformly; namely, it converges under the condition N -1 = o ( e ν ) , where ν = ν ( K ) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final K th iteration, the difference scheme converges e -uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known e -uniformly convergent schemes on piecewise uniform grids. DOI: 10.1134/S0965542508060080

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an e -uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an e -uniform bound O ( + ) is obtained, where δ 1 and δ 0 are the error components due to the approximation of the derivatives with respect to x and t , respectively. Thus, this scheme is e -uni- formly well-conditioned. For the condition number of the scheme on a uniform grid, we have the esti- mate O ( + ) ; this scheme is not e -uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as e 0 , which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O ( + ) . Neither the matrix of the e - uniformly convergent scheme nor the matrix of the scheme on a uniform grid is e -uniformly well-con- ditioned. DOI: 10.1134/S0965542508050072

Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter ɛ 2, ɛ e (0, 1]. For small values of the parameter ɛ, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width ɛ), respectively, which have bounded smoothness for fixed values of the parameter ɛ. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge ɛ-uniformly with the second order of accuracy in x and the first order of accuracy in t, up to logarithmic factors.

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).

Using the technique of majorant functions in approximation of a singular perturbed parabolic convection–diffusion equation on adaptive grids

A grid approximation of the Dirichlet problem is considered on a segment for a parabolic convection-diffusion equation; the high derivative of the equation contains a parameter ε taking arbitrary values from the half-interval (0,1]. A difference scheme on a posteriori adaptive grids is constructed for the boundary value problem. The classic approximation of the equation on uniform grids in the main domain and also in domains refined to improve the accuracy of the grid solution are used. Such subdomains are determined based on majorant functions for singular components of the solutions to the boundary value problem and the difference scheme. Special schemes on a posteriori piecewise-uniform grids are constructed, these schemes permit to obtain approximations convergent in the whole grid domain 'almost e-uniformly', namely, with an error weakly depending on the value of the parameter ε; the scheme converges ε-uniformly with the first order of accuracy (up to logarithmic cofactors) outside a sufficiently small neighborhood of the 'outlet' part of the boundary through which the characteristics of the limit equation leave the domain.

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ɛ taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ɛ-uniformly, i.e., with an error that weakly depends on the parameter ɛ: |u(x, t) − z(x, t)| ≤ M[N1−1 ln2N1 + N0−1 lnN0 + ɛ−1N1−K lnK−1N1], (x, t) e Ḡh, where N1 + 1 and N0 + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ɛ-uniformly at a rate O(N1−1 ln2N1 + N0−1 lnN0), where σ ≤ MN1−K + 1 lnK−1N1 for K ≥ 2.

Optimal piecewise uniform grids for singularly perturbed equations of a convection-diffusion type

We consider the grid approximation of the Dirichlet problem for equations of an elliptic type in a strip. The higher derivatives of the equations contain the parameter e that takes arbitrary values within the half-interval (0, 1]. These convection-diffusion equations degenerate for e =0 into first-order equations. For the small values of the parameter boundary layers may occur in a neighbourhood of the part of the boundary through which the characteristics of the degenerate equations leave the domain. A special (monotonic) difference scheme on piecewise uniform grids which are refined in the s-neighbourhood of the boundary layer is used for the discretization of this problem. The scheme converges e-uniformly with rate O((N 11 lnN1+N1 2 )); N1+1 and N2 +1 are the number of nodes on a rectangular mesh across the strip and along the strip on a unit segment, respectively. It is shown that on the family of piecewise uniform grids (specified by the value s and the node density in a neighbourhood of the boundary layer) this convergence rate as to the error estimate obtained) is unimprovable in the case of the classical difference approximations of a boundary value problem. We find the grid parameters of this family, which allow us (for the same convergence rate) to decrease the error of an approximate solution. For these optimal grids the ratio of the number of nodes of the grid w1 (the grid across the strip), which are located in the s-neighbourhood of the boundary layer, to the total number of nodes of this grid tends to unity as N1 -> ∞