Search

The paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L 2 0 , 1 . We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods.

We study the transformation of the vertices of a certain simple simplicial complex in n -dimensional Euclidian space and prove that the resulting set of simplices is a simplicial complex with an approximate rotational symmetry. Such simplicial complexes have applications in computing Lyapunov function for nonlinear dynamical systems using linear optimization and are also of interest for other applications.

This study relates to nonzero-sum stochastic games with perfect information. It proposes an efficient combined Chebyshev spectral collocation method (CSCM) with the policy iteration (PI) algorithm for solving nonlinear coupled Hamilton–Jacobi (HJ) equations. The proposed approach is comprised of two steps. First, the PI algorithm is used to reduce the nonlinear coupled HJ equations to a sequence of linear uncoupled PDEs. Then, these equations are approximated by the CSCM. The CSCM＋PI is especially useful when the CSCM fails due to the increasing number of collocation points for solving the associated system of nonlinear algebraic equations. The main advantage of the resulting method is that it converts nonlinear coupled HJ equations to the systems of linear algebraic equations, which can be solved readily. Convergence analysis of this method is also provided in detail. To confirm the accuracy and validity of the proposed computational algorithm, several illustrative examples are presented.

In this paper, a second order accurate augmented matched interface and boundary (MIB) is introduced for solving two-dimensional (2D) elliptic interface problems with piecewise constant coefficients. The augmented MIB seamlessly combines several key ingredients of the standard MIB, augmented immersed interface method (IIM), and explicit jump IIM, to produce a new fast interface algorithm. Based on the MIB, zeroth and first order jump conditions are enforced across an arbitrarily curved interface, which yields fictitious values on Cartesian nodes near the interface. By using such fictitious values, a simple procedure is proposed to reconstruct Cartesian derivative jumps as auxiliary variables and couple them with the jump-corrected Taylor series expansions, which allow us to restore the order of the central difference across the interface to two. Moreover, by using the Schur complement to disassociate the algebraic computation of auxiliary variables and function values, the discrete Laplacian can be efficiently inverted by using the fast Fourier transform (FFT). It is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size. As a consequence, the total computational cost of the augmented MIB is about O ( n 2 log n ) for a Cartesian grid with dimension n × n in 2D. Therefore, the augmented MIB outperforms the classical MIB in all cases by significantly reducing the CPU time, while keeping the same second order of accuracy in dealing with complicated interfaces.

We propose an error-adaptive method for constructing time-series models for novelty detection using radial basis functions. The optimization problem is implemented in an incremental, or online manner, and permits the balancing of the quality of the data fit and the sparsity of the model in the objective function of a linear program. The resulting algorithm is applied to several examples of streaming time-series data. Novel data points are identified using an infeasibility criterion while feasible data points are not used for training and may be discarded. In practice, only a small fraction of observed streaming data points are actually required to update the model. The sparsity promoting term in the objective function serves to determine the location and number of RBFs required to fit the data. Balancing sparsity with the accuracy of the model allows the user to adjust model complexity. It appears that the ability to adapt the error bound in the optimization problem may help to prevent over-fitting. We illustrate the approach with several examples and show that the fitting procedure is robust to additive Gaussian noise with non-stationary variance. We compare the proposed algorithm with several other methods in the literature, including both online and batch algorithms.

Generalized Rayleigh distribution is an important continuous life distribution, which plays a key role in reliability engineering research. In this paper, the statistical inference of reliability of Generalized Rayleigh distribution is studied based on the progressively type-II censored data. First we obtain the estimates through maximum likelihood estimation. And by using Bootstrap methods (Boot-p and Boot-t), we estimate the confidence intervals of parameters and reliability. In addition, based on the square error loss function, the Bayes estimation of the reliability function is carried out by adopting the Gibbs sampling method, and the point estimation, the highest posterior density ( H P D ) confidence interval estimation are carried out respectively. A numerical simulation and an authentic example are used to illustrate the proposed inference methods. Finally, the approximate estimation of stress–strength reliability is further studied.

The short pulse equation was introduced by Schafer–Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schrodinger equation does not possess, have drawn much attention. In such a region, the solutions can become quite close to a singularity, and thus existing numerical methods cease to work stably, or even if they do, they require high computational cost. In this paper, we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure-preserving methods. The resulting scheme successfully works even when the singularity occurs, and thus gives a highly robust method for resolving near singular solutions of interest. It is confirmed by numerical experiments, and some new insights about derivative blow-up are obtained.

In this paper, we consider the estimates for the scale parameter of an exponentiated half logistic distribution under progressively Type-II censoring and general progressively Type-II censoring, respectively. Because of the complex form of the traditional maximum likelihood estimator, it is hard to get an explicit solution. Therefore, we propose three new estimation methods based on censored data to obtain approximate estimates of the scale parameter. The first two are approximate maximum likelihood estimation methods based on Taylor expansion and the third is the approximate estimation method based on pivotal quantities we establish. Furthermore, we consider asymptotic confidence intervals based on pivotal quantities. Finally, simulation studies are conducted to compare the performance of the proposed estimators. A real dataset analysis is carried out to illustrate the proposed methods.

We study a quasi-one-dimensional steady-state Poisson–Nernst–Planck model with a local hard-sphere potential for ionic flows of two oppositely charged ion species through a membrane channel. Of particular interest are qualitative properties of ionic flows in terms of individual fluxes J k without assuming electroneutrality boundary conditions. For the first order approximation J k 1 (in diameter of charged particle) of J k , our result shows that the quantities ∂ V J k 1 and ∂ V λ 2 J k 1 that play critical roles in characterizing ion size effects on ionic flows can be both negative depending sensitively on boundary concentrations and relative ion valences while they are always positive under electroneutrality conditions. This new observation indicates that, for fixed ion species, opposite finite size effects on ionic flows can occur determined further by boundary concentrations. Numerical simulation is performed to detect some critical potentials closely related to ion size effects. Numerical simulation results are consistent with our analytical predictions.

In this paper we propose and analyze an energy stable numerical scheme for the Cahn–Hilliard equation, with second order accuracy in time and the fourth order finite difference approximation in space. In particular, the truncation error for the long stencil fourth order finite difference approximation, over a uniform numerical grid with a periodic boundary condition, is analyzed, via the help of discrete Fourier analysis instead of the standard Taylor expansion. This in turn results in a reduced regularity requirement for the test function. In the temporal approximation, we apply a second order BDF stencil, combined with a second order extrapolation formula applied to the concave diffusion term, as well as a second order artificial Douglas–Dupont regularization term, for the sake of energy stability. As a result, the unique solvability, energy stability are established for the proposed numerical scheme, and an optimal rate convergence analysis is derived in the l ∞ ( 0 , T ; l 2 ) ∩ l 2 ( 0 , T ; H h 2 ) norm. A few numerical experiments are presented, which confirm the robustness and accuracy of the proposed scheme.

In this paper, we propose two numerical methods for solving the space-fractional reaction–diffusion equations, which are based on a Fourier spectral approach in space and exponential time differencing schemes in time. The advantages of the approaches are that they attain spectral convergence, produce a full diagonal representation of the fractional operator, and the extension to multiple spatial dimensions is the same as the one-dimensional space. That can overcome the constraints associated with many of the numerical schemes for these equations such as the computational efficiency caused by the non-locality of the fractional operator, which results in full, dense matrices. Moreover, the proposed schemes are second-order convergent, unconditionally stable, and highly efficient due to the predictor–corrector feature when comparing them with the existing method. It is observed that the scheme based on using Pade(1,1) approximations to the matrix exponential function introduces oscillations with non-smooth initial data for some time steps due to high frequency components present in the solution which diminish as the fractional order decreases. However, the scheme based on Pade(0,2) approximations to the matrix exponential function is oscillation-free for any time step. Numerical experiments for well-known models from the literature are performed to show the reliability and effectiveness of the proposed methods.

In this note, we consider a general 2 × 2 system of nonlinear Volterra type integral equations. The modified Newton method (modified NM) is used to reduce the nonlinear problems into 2 × 2 linear system of algebraic integral equations of Volterra type. The latter equation is solved by discretization method. Nystrom method with Gauss–Legendre quadrature is applied for the kernel integrals and Newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points. Existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained. Finally, numerical examples are provided to show the validity and efficiency of the method presented. Numerical results reveal that the proposed methods is efficient and accurate. Comparisons with other methods for the same problem are also presented.

The computational analysis of the Cauchy problem for semi-linear Klein-Gordon equations in the de Sitter spacetime is considered. Several simulations are performed to show the time-global behaviors of the solutions of the equations in the spacetime based on the structure-preserving scheme. It is remarked that the sufficiently large Hubble constant yields the strong diffusion-effect which gives the long and stable simulations for the defocusing semi-linear terms. The reliability of the simulations is confirmed by the preservation of the numerically modified Hamiltonian of the equations., 25 pages, 18 figures

In this paper, we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods. It is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or Lyapunov functions. We also consider order estimates of the new methods. Furthermore, we explore and discuss the application of our methods in important stiff gradient systems, and it turns out that our methods are unconditionally energy-diminishing and strongly damped even for very stiff gradient systems. Practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear Schrodinger equation. As a byproduct of this paper, arbitrary-order trigonometric/RKN collocation methods are also presented and analysed for second-order highly oscillatory/general systems. The paper is accompanied by numerical results that demonstrate the great potential of this work.

A general framework is given for applying the Newton–Raphson method to solve power flow problems, using power and current-mismatch functions in polar, Cartesian coordinates and complex form. These two mismatch functions and three coordinates, result in six possible ways to apply the Newton–Raphson method for the solution of power flow problems. We present a theoretical framework to analyze these variants for load (PQ)buses and generator (PV)buses. Furthermore, we compare newly developed versions in this paper with existing variants of the Newton power flow method. The convergence behavior of all methods is investigated by numerical experiments on transmission and distribution networks. We conclude that variants using the polar current-mismatch and Cartesian current-mismatch functions that are developed in this paper, performed the best result for both distribution and transmission networks.

Geometrical symmetry is found in many linear field problems. Group representation theory is the only valuable tool for exploiting this property from a computational point of view. In this context, here we apply a technique for taking into account equivariance in the numerical treatment of space–time boundary integral equations, which are invariant under a finite group G of congruences of R 3 , each related to one of the so-called Platonic solids. This technique is based upon suitable restriction matrices strictly related to a system of unitary, irreducible, pairwise not-equivalent matrix representations of G . The main development is expounded in the framework of space–time energetic boundary element method applied to Neumann exterior wave propagation problems, where the discretization matrices have a block lower triangular Toeplitz structure, and the diagonal block, to be inverted at each time step, is typically dense. Several numerical results, related to computational saving, are presented.

In this paper, we propose and analyze a conservative high-order compact finite difference scheme for the Klein–Gordon–Schrodinger (KGS) equation with Dirichlet boundary condition. By introducing the time-derivative of one solution as a separate dependent variable, we rewrite the KGS equation as a system of partial differential equations (PDEs) and obtain a two-level compact finite difference scheme. Based on the energy method and matrix knowledge on the vector form, we convert the point-wise form of the proposed compact scheme into an equivalent vector form and analyze its conservative and convergence properties. We prove that the proposed scheme preserves the total energy and charge in the discrete forms and the convergence rate of the scheme, without any restrictions on the grid ratio, is at the order of O ( h 4 + τ 2 ) in L 2 -norm, where h and τ are spatial and temporal steps, respectively. Numerical experiments are carried out to verify the performance of the scheme.

This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O ( N 3 ) , where N is the number of mesh-points, to only O ( N ) .

A new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the solution of fourth order Sturm--Liouville problem on a finite interval in the Hilbert space. The eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated. The sufficient conditions of an exponential convergence rate of the proposed approach are received. The obtained estimates of the absolute errors of FD-method significantly improve the accuracy of the estimates obtained earlier by I.P~Gavrilyuk, V.L.~Makarov and A.M.~Popov in 2010. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential coefficients and the correction number. Our method uses only the algebraic operations and basic operations on $(2\times 1)$ column vectors and $(2\times 2)$ matrices. The proposed approach does not require solving any boundary value problems and computations of any integrals, unlike the previous variants of FD-method by I.P.~Gavrilyuk, V.L.~Makarov, A.M.~Popov and N.M.~Romaniuk in 2010 and 2017. The corrections to eigenpairs are computed exactly as analytical expressions, and there are no rounding errors. The numerical examples illustrate the theoretical results. The numerical results obtained with the FD-method are compared with the numerical test results obtained with other existing numerical techniques., 38 pages, 3 figures, 5 tables, 3 appendices

We consider the inverse problem of determining an unknown vectorial source current distribution associated with the homogeneous Maxwell system. We propose a novel non-iterative reconstruction method for solving the aforementioned inverse problem from multi-frequency far-field measurements. The method is based on recovering the Fourier coefficients of the unknown source. A key ingredient of the method is to establish the relationship between the Fourier coefficients and the multi-frequency far-field data. Uniqueness and stability results are established for the proposed reconstruction method. Numerical experiments are presented to illustrate the effectiveness and efficiency of the method.

In this paper, we obtain a ( p , v ) -extension of the Appell hypergeometric function F 1 ( ⋅ ) , together with its integral representation, by using the extended Beta function B p , v ( x , y ) introduced in Parmar et al. (2015). Also, we give some of its main properties, namely the Mellin transform, a differential formula, recursion formulas and a bounded inequality. In addition, some new integral representations of the extended Appell function F 1 , p , v ( ⋅ ) involving Meijer’s G -function are obtained.

In this paper we propose and study a new Lagrange finite element method for the two-dimensional Maxwell’s equations. Its solution may be singular because of the nonsmooth domain with reentrant corners. The proposed method allows the standard Lagrange elements on barycentric refinements of any order greater than or equal to two. We analyze the proposed method for the eigenvalue problem and the indefinite source problem, obtaining the well-posedness and optimal error estimates. Numerical results are presented for confirming the theoretical results.

In this paper, we numerically simulate the flow of blood in two benchmark problems: the flow in a sudden expansion channel and the flow through an idealized curved coronary artery with pulsatile inlet velocity. Blood is modeled as a suspension (a non-linear complex fluid) and the movement of the red blood cell (RBCs) is modeled by using a concentration flux equation. The viscosity of blood is obtained from experimental data. In the sudden expansion flow, the predicted velocity profiles for two different Reynolds numbers (based on the inlet velocity) agree well with the available experiments; furthermore, the numerical results also show that after the sudden expansion there exists a RBCs depletion region. For the second problem, the idealized curved coronary artery, it is found that the RBCs move towards and concentrate near the inner surface where the viscosity is higher and the shear stress is lower; this phenomenon may be related to the atherosclerotic plaque formation which usually occurs on the inside surface of the arteries.

A novel mathematical framework is derived for the addition of nodes to univariate and interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde matrix describing the relation between the nodes and the weights and can be used to determine all nodes that can be added to an interpolatory quadrature rule with positive weights such that the positive weights are preserved. In the case of addition of a single node, the derived inequalities that describe the regions where nodes can be added are explicit. Besides addition of nodes these inequalities also yield an algorithmic description of the replacement and removal of nodes. It is shown that it is not always possible to add a single node while preserving positive weights. On the other hand, addition of multiple nodes and preservation of positive weights is always possible, although the minimum number of nodes that need to be added can be as large as the number of nodes of the quadrature rule. In case of addition of multiple nodes the inequalities describing the regions where nodes can be added become implicit. It is shown that the well-known Patterson extension of quadrature rules is a special case that forms the boundary of these regions and various examples of the applicability of the framework are discussed. By exploiting the framework, two new sets of quadrature rules are proposed. Their performance is compared with the well-known Gaussian and Clenshaw–Curtis quadrature rules, demonstrating the advantages of our proposed nested quadrature rules with positive weights and fine granularity.

In this paper, a class of new extended shift-splitting (NESS) preconditioners containing t > 0 are proposed for saddle point problems, which contain the existing shift-splitting-based preconditioners as special cases. The convergence properties of the NESS iteration and the spectral distribution of the NESS preconditioned matrix are investigated. We give an estimated bound for the real eigenvalues of the NESS preconditioned matrix whose left end is positive and show that the non-real eigenvalues of the NESS preconditioned matrix are located in an intersection of two rings, particularly, these non-real eigenvalues are located in an intersection of two rings and one circle if t ≥ 1 2 . Moreover, we show that under appropriate conditions our proposed eigenvalue bounds are tighter than the corresponding bounds given in Ren et al. (2017). Using Fourier analysis, we derive an optimized parameter t ∗ independent of the viscosity ν for the continuous version of the NESS preconditioned GMRES method for the 2D Stokes equation. Moreover, we find that the NESS preconditioned GMRES method with a constant multiple of the optimized parameter t ∗ is effective and robust to solve 2D Stokes problems in practice. Numerical experiments are used to verify our theoretical results.

In this paper, we consider the problem of pricing European options, namely vanilla options, binary options and exchange options, whose underlying assets prices dynamics follow Markovian regime switching exponential Levy models with stochastic interest rates, where the stochastic interest rates are driven by Markovian regime switching Hull–White process. We obtain the integral representations of the option prices by Fourier transform (FT) technique and some numerical results of 3-state case for Merton jump-diffusion model by the fast Fourier transform (FFT) approach. The numerical results show that the pricing formulas are considerably accurate and easy to be implemented.

Phase retrieval (PR) problem is an inverse problem which arises in various applications. Based on the Wirtinger flow method, an algorithm utilizing the sparsity priority called SWF (Sparse Wirtinger Flow) is proposed in this paper to deal with the PR problem. Firstly, the support of the signal is estimated besides the initialization is evaluated based on this support. Then the evaluation is updated by the hard-thresholding method from this initialization. We prove that for any k -sparse signal with length n , SWF has a linear convergence with O ( k 2 log n ) phaseless Gaussian random measurements. To get e accuracy, the computational complexity of SWF is O ( k 3 n log n log 1 e ) . Numerical tests also demonstrate that SWF has a higher recovery rate than other algorithms compared especially when we have no prior information about sparsity k . Moreover, SWF is robust to the noise.

In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with default risks and jumps for an insurer whose objective is to maximize the expected exponential utility of terminal wealth. The surplus process of the insurer is assumed to follow a Brownian motion with drift (which is an approximation of the classical compound Poisson risk model). Assume that the insurer is allowed to purchase proportional reinsurance and invest in a financial market which consists of a risk-free asset, a defaultable bond and a risky asset whose price process is governed by a jump–diffusion model. Using stochastic control approach, we establish the robust Hamilton–Jacobi–Bellman equations for the post-default case and the pre-default case, respectively. Furthermore, we derive the closed-form expressions of the optimal strategies and the corresponding value functions in both cases. Finally, we provide numerical examples to illustrate the effects of some model parameters on the robust optimal strategies.

In this paper, the distributed Lagrange multiplier/fictitious domain (DLM/FD) finite element method is studied for a generic Stokes/elliptic interface problem with jump coefficients which belongs to a type of linearized stationary fluid–structure interaction problem. A mixed finite element discretization is developed for the proposed DLM/FD method for Stokes/elliptic interface problem and analyzed on the aspects of well-posedness, stability and optimal convergence. Numerical experiments are carried out and the theoretical error estimates of DLM/FD finite element method are validated.

In this paper, we establish two different α th moment ψ -type stability criterions on the exact and numerical solutions of the stochastic pantograph differential equations. One is established by the virtue of Lyapunov method, and the other is used by the continuous and discrete Razumikhin technique. By comparing the two stability criterions, we can easily see that the conditions constructed by the Razumikhin-type technique are better than those constructed by the virtue of Lyapunov method. Using the conditions constructed for the α th moment ψ -type stability, we study the stability of the Euler–Maruyama method and the backward Euler–Maruyama method, respectively, for a special class of the stochastic pantograph differential equations. Finally, examples are given to illustrate the consistence with the theoretical results on the α th moment ψ -type stability.

In this paper, we present L 2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrodinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Suli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrodinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k + 1 , whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2 k . Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O ( h k + 1 ) to at least O ( h 2 k + 1 ) for Schrodinger equations by using alternating numerical fluxes.

We develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span. The schemes yield two extra orders of convergence (superconvergence) on the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. Both dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. The resulting schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems. We also derive an a posteriori error estimator for the eigenvalue error based on the superconvergence result. We verify with numerical examples the analysis of the performance of the proposed schemes.

Prior work on computable defect-based local error estimators for (linear) time-reversible integrators is extended to nonlinear and nonautonomous evolution equations. We prove that the asymptotic results from the linear case [W. Auzinger and O. Koch, An improved local error estimator for symmetric time-stepping schemes, Appl.Math.Lett. 82 (2018), pp. 106-110] remain valid, i.e., the modified estimators yield an improved asymptotic order as the step size goes to zero. Typically, the computational effort is only slightly higher than for conventional defect-based estimators, and it may even be lower in some cases. We illustrate this by some examples and present numerical results for evolution equations of Schr\"odinger type, solved by either time-splitting or Magnus-type integrators. Finally, we demonstrate that adaptive time-stepping schemes can be successfully based on our local error estimators.

Commonly used diagonal pivoting strategies for general symmetric matrix would bring about triangular factorization PAP’=LDL’, in which D is block diagonal matrix with either 1 × 1 or 2 × 2 diagonal blocks. This paper proposed a modified add–sub pivoting strategy resulting in D with no 2 × 2 diagonal block and a corresponding symmetric triangular factorization algorithm for solving symmetric linear system Ax=b. The factorization algorithm is based on symmetric Gaussian elimination; The pivoting strategy involves searching two suitable columns in the trailing submatrix A k by rook method, interchanging the first one of them with column k, and then adding the second one multiplied by a scalar γ to column k. This strategy is similar to an earlier one proposed by Dax; Its comparisons required are still O(n2) ∼ O(n3), but the bound on elements of L is lower, and especially the growth factor is bounded by 3. The results from numerical experiments with both ill-conditioned and good-conditioned symmetric indefinite linear systems show that the new algorithm is competitive with the B–K algorithm, the BBK algorithm, and the LU factorization with partial pivoting.

In this paper, a Crank–Nicolson difference scheme is first derived for solving the nonlinear time–space fractional Schrodinger equations. The truncation error and stability of the scheme are discussed in detail. The existence of the numerical solution is shown by the Brouwer fixed point theorem. For improving the calculating efficiency, a three-level linearized difference scheme is also proposed and analyzed. Both schemes are subsequently extended to the nonlinear coupled equations, and some similar results are given and proved. Several numerical experiments are included to verify the accuracy and efficiency of the two types of schemes, and comparison with the related work is presented.

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (PDEs): wave propagation, advection–diffusion, plates, beams, etc. The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear parabolic PDEs. This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the pde’s are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Criteria for convergence of fully explicit method using GFDM for different non linear parabolic PDEs are given. This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.

This paper concerns numerical stabilities of a decomposed compact finite difference method for solving Helmholtz partial differential equation problems with large wave numbers. Radially symmetric transverse fields and standard polar coordinates are considered. A decomposition is implemented to remove the anticipated singularity in the transverse direction. Compact scheme structures are introduced in the transverse direction to raise the accuracy for laser and subwavelength optical computations. It is proven that, while the highly accurate compact algorithm shies away from the conventional stability in the von Neumann sense, it is asymptotically stable with index one. Additionally, a discussion on theoretical and numerical stabilities for this method is presented. Numerical experiments further demonstrate the high reliability of the scheme when implemented in highly oscillatory optical self-focusing wave propagation simulations.

The article of record as published may be found at https://doi.org/10.1016/j.cam.2018.12.044 The Karush–Kuhn–Tucker conditions for a given nonlinear programming problem are generated as the transversality conditions of an optimal control problem. The directional derivatives of the objective- and constraint-functions supply the vector fields for the optimal control problem with the search vector as the control variable. Zero-Hamiltonian trajectories along the steepest descent of control Lyapunov functions provide optimal optimization algorithms. The optimality of the algorithm also depends upon the choice of a metric for the finite dimensional control space. Many well-known algorithms – such as Newton’s method, the first-order Lagrangian method, the steepest descent method and Richardson’s method, to name a few – are derived by minimizing the Lie derivative of a quadratic control Lyapunov function. Merit functions in optimization may also be generated using the concept of a control Lyapunov function. These results suggest that optimal control principles hold the potential for a unified theory for optimization.

In this paper, we present a new four-step implicit P-stable Obrechkoff method of tenth algebraic order for solving one-dimensional second-order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable Schrodinger equation. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.

International audience; This paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.

This article proposes a generalization of the Fourier interpolation formula, where a wider range of the basic trigonometric functions is considered. The extension of the procedure is done in two ways: adding an exponent to the maps involved, and considering a family of fractal functions that contains the standard case. The studied interpolation converges for every continuous function, for a large range of the nodal mappings chosen. The error of interpolation is bounded in two ways: one theorem studies the convergence for Holder continuous functions and other develops the case of merely continuous maps. The stability of the approximation procedure is proved as well.

In this work, we study some numerical properties of the continuous Newton’s method, the continuous version of the classical Newton’s method for solving nonlinear equations p ( z ) = 0 . In fact, continuous Newton’s method is an initial value problem whose solutions flow to a root of the equation. We show the influence of the multiplicity of the roots of the considered equation in the Jacobian matrix related to the problem. In addition, we study some modifications of the continuous Newton’s method that allow us to increase the velocity of the convergence of the solutions towards the roots of p ( z ) = 0 .

We use an auxiliary point on the semilocal convergence of Newton’s method when the majorant principle of Kantorovich is applied to operators with high order derivatives satisfying a center Lipschitz type condition, so that we extend the classical conditions of these types, that are centered at the starting point of Newton’s method, to other points belonging to the domain of definition of the operator involved. This extension provides a modification of the domain of starting points for Newton’s method which allows increasing the choice of starting points. We illustrate this study with nonlinear Fredholm integral equations.

In this paper some properties of two-determinant pivoting for Neville elimination are presented. In particular, we consider a zero-increasing property and we show an optimal normwise growth factor. Comparisons with other pivoting strategies for Neville elimination and with Gaussian elimination with partial pivoting of almost strictly sign regular matrices are performed. Numerical examples are included.

In this paper we propose an algorithm to numerically simulate Markov processes of jump type. While these processes can naturally be generated as solutions to jump-SDEs the algorithm we propose is instead based on the interlacing construction of the process. We show that one can construct in the sense of strong convergence a high order numerical scheme based on high order ODE solvers. This result is in sharp contrast to the well known difficulty of constructing high-order numerical schemes for diffusion processes.

A reduced/selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity. Numerical experiments show that, for the iso-parametric biquadratic finite element approximation of the displacement field, the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field, if and only if the reduced integration is applied to the bulk energy term using no more than 2 Gaussian quadrature nodes in each dimension.

We analyze a fully discrete numerical scheme for the model describing two-phase immiscible flow in porous media with dynamic effects in the capillary pressure. We employ the Euler implicit method for the time discretization. The spatial discretization is based on the mixed finite element method (MFEM). Specifically, the lowest order Raviart–Thomas elements are applied. In this paper, the error estimates for the saturation, fluxes and phase pressures in L ∞ ( 0 , T ; L 2 ( Ω ) ) are derived for the temporal and spatial discretization to show the convergence of the scheme. Finally, we present some numerical results to support the theoretical findings.

In this paper we study the efficient solution of the well-known Korteweg–de Vries equation, equipped with periodic boundary conditions. A Fourier–Galerkin space semi-discretization at first provides a large-size Hamiltonian ODE problem, whose solution in time is then carried out by means of energy-conserving methods in the HBVM class (Hamiltonian Boundary Value Methods). The efficient implementation of the methods for the resulting problem is also considered and several numerical examples are reported.

A method for solving boundary value problems usually referred to as the unified transform or the Fokas method was introduced in the late nineties. A crucial role in this method is played by the so-called global relation, which characterizes the associated generalized Dirichlet-to-Neumann map, which can be used to determine the unknown boundary values in terms of the given boundary data. Then, the solution can be expressed via an integral formulated in the complex Fourier plane. This method can be considered as the spectral analogue of the classical Boundary Integral Method, which is formulated in the physical plane. Recently, it has been realized that the numerical implementation of the unified transform leads to a collocation-type method: this involves expanding the unknown boundary values in terms of appropriate basis functions and choosing a suitable set of complex collocation points. Herewith, an iterative spatial-stepping algorithm for computing the solution of a linear PDE in the interior of a convex polygon, is presented. The starting point of this algorithm is the analysis of the approximate global relation and its numerical solution. In more details, the interior of the polygon is partitioned into smaller concentric polygonal domains where the solution is computed recursively starting from the boundaries and proceeding towards the center. The solution of each inner polygon is computed by a spatial-stepping scheme using the Dirichlet and Neumann values of the respective outer polygon, which are computed via the approximate global relation.

In this study, an operational approach, based on the shifted Jacobi polynomials, is presented to solve a class of weakly singular fractional integro-differential equations. The fractional derivative operators are considered as the Caputo sense. In addition to finding the operational matrices of integration and product, a new operational matrix is derived to be approximated the singular kernels of this class of the functional equations. These operational matrices together with the collocation method are applied to convert the main equation into a linear or non-linear system of algebraic equations. By presenting some theorems, conditions of the existence and uniqueness of the solution of the main equation and its corresponding algebraic system are investigated. Also, the stability and convergence of the proposed method are studied and an error bound is obtained in a Jacobi-weighted Sobolev space. Finally, some illustrative examples are provided to demonstrate the efficiency and simplicity of the suggested approach.