Order Of Convergence Research Articles

Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂t−i△Γ where △Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

Higher-order Newton methods with polynomial work per iteration

We present generalizations of Newton's method that incorporate derivatives of an arbitrary order d but maintain a polynomial dependence on dimension in their cost per iteration. At each step, our dth-order method uses semidefinite programming to construct and minimize a sum of squares-convex approximation to the dth-order Taylor expansion of the function we wish to minimize. We prove that our dth-order method has local convergence of order d. This results in lower oracle complexity compared to the classical Newton method. We show on numerical examples that basins of attraction around local minima can get larger as d increases. Under additional assumptions, we present a modified algorithm, again with polynomial cost per iteration, which is globally convergent and has local convergence of order d.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

Derivative free processes of high order for nondifferentiable equations in Banach spaces

In this article, we consider two parametric classes of derivative free iterative methods of high order of convergence in Banach spaces. We study local and semilocal convergence results in Banach spaces for both families of iterative processes. We carry out a numerical application to solve a nonlinear and nondifferentiable Fredholm integral equation in the Banach space of continuous functions with the maximum norm.

A resolution-enhanced seventh-order weighted essentially non-oscillatory scheme based on non-polynomial reconstructions for solving hyperbolic conservation laws

In high-resolution numerical simulations of flows characterized by both multiscale turbulence and discontinuities, the conflict between spectral characteristics and stability becomes increasingly pronounced as the order of accuracy improves. To address this challenge, we proposed a novel seventh-order weighted essentially non-oscillatory scheme (WENO-K7). This scheme utilizes non-polynomial reconstructions by incorporating kriging interpolation and Gaussian exponential function. Then, a hyper-parameter associated with the Gaussian function is adaptively optimized to achieve higher convergence orders on sub-stencils, reducing numerical errors on global stencils. Additionally, a criterion based on monotone interpolations is devised to automatically identify problematic hyper-parameters, facilitating the transition from non-polynomial to polynomial reconstructions near discontinuities and preserving the essentially non-oscillatory property. Compared to the conventional seventh-order WENO-Z7 scheme, WENO-K7 scheme exhibits smaller computational error and reduced numerical dissipation in smooth regions while maintaining non-oscillatory and high-resolution capabilities around discontinuities. Results from various one- and two-dimensional benchmark cases demonstrate that the proposed WENO-K7 scheme outperforms the widely used WENO-Z7 scheme with only a 12% increase in computational cost. Moreover, the WENO-K7 scheme shares the same sub-stencils as the WENO-Z7 scheme, making it easily applicable to other variants of seventh-order WENO schemes and enhancing their spectral characteristics.

Almost lacunary statistical and strongly almost lacunary convergence of order (β,γ) of sequences of fuzzy numbers

The main purpose of this article is to introduce the concepts of almost lacunary statistical convergence and strongly almost lacunary convergence of order (β,γ) of sequences of fuzzy numbers with respect to an Orlicz function. We give some relations between strongly almost lacunary convergence and almost lacunary statistical convergence of order (β,γ) of sequences of fuzzy numbers, where β and γ are two fixed real numbers such that 0 β≤γ≤1.

On high-order schemes for the space-fractional conservative Allen–Cahn equations with local and local–nonlocal operators

In this study, we focus on two fractional conservative Allen–Cahn equations with a nonlocal space-independent operator (called the RSLM operator) and a local–nonlocal space–time dependent operator (called the BBLM operator), respectively. Recently, scholars have found that the fractional Allen–Cahn equation is better than the classical equation for describing the interface thickness. Subsequently, the fractional conservative Allen–Cahn equation was used for conservative fluid dynamics. We find that two fractional conservative Allen–Cahn equations both hold the mass conservation, energy stability, and maximum bound principle. In this study, we construct second- and third-order numerical algorithms for two fractional conservative Allen–Cahn equations, and prove the mass conservation and energy decrease laws for them. To our knowledge, for the conservative Allen–Cahn equation, there are only a few works for high-order schemes with proof of their energy stability. We then present several numerical examples whose results show that the convergence orders of the numerical solutions reach the second- and third-order in time as expected. We also simulate some phase transition behaviors to be mass conservative and energy stable, as described by the fractional conservative Allen–Cahn equations, and have some interesting discoveries. It is found that the BBLM formulation can better maintain small features, but the RSLM formulation is better at capturing the profile of the interface. We also compare two fractional conservation equations with the fractional Cahn–Hilliard equation, whose changes are similar to those of the RSLM formulation. In particular, we propose the variable-step BDF2 and BDF3 methods, and prove the mass conservation and energy stability under a time-ratio constraint.

Determination of an unknown coefficient in the Korteweg–de Vries equation

Abstract In this paper, a space-time spectral method for solving an inverse problem in the Korteweg–de Vries equation is considered. Optimal order of convergence of the semi-discrete method is obtained in L 2 {L^{2}} -norm. The discrete schemes of the method are based on the modified Fourier pseudospectral method in spatial direction and the Legendre-tau method in temporal direction. The nonlinear term is computed via the fast Fourier transform and fast Legendre transform. The method is implemented by the explicit-implicit iterative method. Numerical results are given to show the accuracy and capability of this space-time spectral method.

Deep learning based on randomized quasi-Monte Carlo method for solving linear Kolmogorov partial differential equation

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations (PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization (ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is O(n−1+ϵ) for arbitrarily small ϵ>0, while for the MC method it is O(n−1/2+ϵ) for arbitrarily small ϵ>0. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as n increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative L2 error than that based on the MC method.

Convergence analysis for minimum action methods coupled with a finite difference method

Abstract The minimum action method (MAM) is an effective approach to numerically solving minima and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method with nonuniform mesh, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are $1/2$ and $1$, respectively. Our main result also reveals the convergence of the stochastic $\theta $-method for SDEs with small noise in terms of large deviations. Numerical experiments are reported to verify the theoretical results.

Solving multi-dimensional European option pricing problems by integrals of the inverse quadratic radial basis function on non-uniform meshes

This paper explores multi-asset options as a means to diversify portfolios, mitigating risk across various assets. We present a numerical method using radial basis function-generated finite difference solvers via integrals of the inverse quadratic kernel. Our method introduces new weights for the task we are dealing with. We derive and compute analytical solutions to approximate function derivatives on three-node stencils with non-uniform and uniform distances. Our findings highlight the convergence order of the proposed analytical weights. Numerical examples illustrate the theory.

Atomicity of Boolean algebras and vector lattices in terms of order convergence

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

A generalized energy and QUadratic Invariant Preserving (GEQUIP) method for Hamiltonian systems with multiple invariants

It is well-known that the symplecticity and energy-conservation are the most characteristic properties of a Hamiltonian system. Meanwhile, many Hamiltonian problems allow the presence of multiple physical invariants, besides energy. This paper deals with a generalization of the EQUIP methods defined in Brugnano et al. [1], aimed at introducing additional parameters to attain further multiple invariants conservation properties, together with the symplecticity of the map. The solvability of the nonlinear system arising from the conservation requirements on multiple invariants is proved and the order of convergence on the method is discussed. Some numerical tests are reported in order to confirm the efficiency of the methods, which shows the good multiple invariants preserving property of the proposed method compared to Gauss collocation methods and EQUIP methods.

A new pair of block techniques for direct integration of third-order singular IVPs

This paper proposes a new pair of block techniques (NPBT) for the direct solution of third-order singular initial-value problems (IVPs). The proposed method uses a polynomial and two intermediate points to approximate the theoretical solution of third-order singular IVPs, resulting in a reasonable approximation within the integration interval. The method's essential features, including stability and convergence order, are analyzed. The proposed NPBT method is improved by using an embedding-like strategy that allows it to be executed in a variable step size mode in order to gain better efficiency. The effectiveness of the proposed method is assessed using various model problems. The approximate solution provided by the proposed NPBT method is more accurate than that of the existing methods utilized for comparison. This efficient solution positions NPBT as a good numerical method for integrating third-order singular IVP models in the fields of applied sciences and engineering.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.

Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations

The symmetric successive overrelaxation (SSOR) and symmetric accelerated overrelaxation (SAOR) are conventional iterative methods for solving linear equations. In this paper, novel approaches are presented by combining a splitting–linearizing method with SSOR and SAOR for solving a system of nonlinear equations. The nonlinear terms are decomposed at two sides through a splitting parameter, which are linearized around the values at the previous step, obtaining a linear equation system at each iteration step. The optimal values of parameters are determined to minimize the reciprocal of the maximal projection, which are sought in preferred ranges using the golden section search algorithm. Numerical tests assess the performance of the developed methods, namely, the optimal splitting symmetric successive over-relaxation (OSSSOR), and the optimal splitting symmetric accelerated over-relaxation (OSSAOR). The chief advantages of the proposed methods are that they do not need to compute the inverse matrix at each iteration step, and the computed orders of convergence by OSSSOR and OSSAOR are between 1.5 and 5.61; they, without needing the inner iterations loop, converge very fast with saving CPU time to find the true solution with a high accuracy.

Decoupling numerical method based on deep neural network for nonlinear degenerate interface problems

Many practical problems, including modeling composite materials, nuclear waste disposal, oil reservoir simulations, and flows in porous medium, commonly involve interface problems. However, the solution to interface problems with discontinuous coefficients of PDEs using fully decoupled numerical methods is challenging. The main objective is to solve the interface problems with fully decoupled numerical methods. This paper proposes an efficient decoupled numerical method for solving degenerate interface problems with double singularities. First, we divide the whole domain into singular and regular subdomains. Then, we use the Deep Neural Network (DNN) to find the solution on the singular subdomain and approximate the solution on the regular subdomain using the finite difference method. The scheme combines the solutions of singular and regular subdomains, which is an exciting idea. The key to the new approach is to split nonlinear degenerate partial differential equations with an interface into two independent boundary value problems based on deep learning. In this way, the expansion of the solution on the singular domain does not contain undetermined parameters, and two independent boundary value problems can be solved with any well-known traditional numerical methods. The main advantage of the proposed scheme is that we not only get the order of convergence of the degenerate interface problems on the whole domain, but we also can calculate VERY BIG jump ratio (such as 1012:1 or 1:1012) for the interface problems including degenerate and non-degenerate cases. Finally, with examples, we demonstrate the efficiency and accuracy of methods for 1 and 2D problems. It is also interesting that the proposed method is valid for the interface problems with degenerate and non-degenerate cases, we show it with some examples.

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.