In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction

In this work, Abel’s integral operator is represented based on Alpert’s multiwavelets as a sparse matrix and then a non-linear Abel’s integral equation of the second kind is solved by multiwavelets Galerkin method. Nonlinearity and singularity make the numerical procedure more challenging. But the proposed scheme overcomes these problems. Convergence analysis is investigated and some numerical examples

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose–Einstein condensates based on Newton iteration scheme. Instead of treating eigenvalue \(\lambda \) and eigenvector u separately, we regard the eigenpair \((\lambda , u)\) as one element in the composite space \({\mathbb {R}} \times H_0^1(\varOmega )\) and then Newton iteration step is adopted for the nonlinear

Linear and nonlinear evolution equations have been formulated to address problems in various fields of science and technology. Recently, methods using an exponential integrator for solving evolution equations, where matrix functions must be computed repeatedly, have been investigated and refined. In this paper, we propose a new method for computing these matrix functions which is called an inexact

In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an r times differentiable function with the assumption that the rth derivative is positive within precision \(\varepsilon >0\) are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses

Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge–Kutta methods, however, relies on a convergence result that requires weakening many of the order conditions, resulting in schemes whose stages must be implemented in a sequential way. In this work, after showing a stronger

We show that integrating a polynomial f of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous related Bombieri polynomials of degree \(j=1,2,\ldots ,t\), each at a unique point \(\varvec{\xi }_j\) of the simplex. This new and very simple formula could be exploited in finite (and extended finite) element methods, as well as in applications where such

In this paper, a piecewise quadratic finite element method on rectangular grids for \(H^1\) problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be \(O(h^2)\) in the energy norm on uniform grids over a convex domain. A lower bound of the \(L^2\)-norm error is also proved, which makes

For an \(m \times n\) matrix A, the mathematical property that the rank of A is equal to r for \(0< r < \min (m,n)\) is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency k in a mathematically rigorous way and using only floating-point arithmetic. Given an integer

The randomized Kaczmarz (RK) method is an iterative method for approximating the least-squares solution of large linear systems of equations. The standard RK method uses sequential updates, making parallel computation difficult. Here, we study a parallel version of RK where a weighted average of independent updates is used. We analyze the convergence of RK with averaging and demonstrate its performance

Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation that must be solved only over the boundary of the domain

Classical a posteriori error analysis for differential equations quantifies the error in a Quantity of Interest which is represented as a bounded linear functional of the solution. In this work we consider a posteriori error estimates of a quantity of interest that cannot be represented in this fashion, namely the time at which a threshold is crossed for the first time. We derive two representations

Evaluating the action of a matrix function on a vector, that is \(x=f({\mathcal {M}})v\), is an ubiquitous task in applications. When \({\mathcal {M}}\) is large, one usually relies on Krylov projection methods. In this paper, we provide effective choices for the poles of the rational Krylov method for approximating x when f(z) is either Cauchy–Stieltjes or Laplace–Stieltjes (or, which is equivalent

In this paper, a new waveform relaxation variant of the Dirichlet–Neumann algorithm is introduced for general parabolic problems as well as for the second-order wave equation for decompositions with multiple subdomains. The method is based on a non-overlapping decomposition of the domain in space, and the iteration involves subdomain solves in space-time with transmission conditions of Dirichlet and

This paper introduces an adaptive preconditioner for iterative solution of sparse linear systems arising from partial differential equations with self-adjoint operators. This preconditioner allows to control the growth rate of a dominant part of the algebraic error within a fixed point iteration scheme. Several numerical results that illustrate the efficiency of this adaptive preconditioner with a

This manuscript presents a fast direct solution technique for solving two dimensional wave scattering problems from quasi-periodic multilayered structures. When the interface geometries are complex, the dominant term in the computational cost of creating the direct solver scales O(NI) where N is the number of discretization points on each interface and I is the number of interfaces. The bulk of the

This paper introduces the Runge–Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical schemes that avoid the severe step size restriction faced by standard explicit integrators. For optimising quadratic and strongly convex functions, this paper proves

The numerical method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex analysis, its use has been restricted to two dimensions (2D). We first explain its loss of accuracy as panels become curved, using a classical complex approximation

We present a new class of asymptotic preserving trigonometric integrators for the quantum Zakharov system. Their convergence holds in the strong quantum regime \(\vartheta = 1\) as well as in the classical regime \(\vartheta \rightarrow 0\) without imposing any step size restrictions. Moreover, the new schemes are asymptotic preserving and converge to the classical Zakharov system in the limit \(\vartheta

The steady state fractional convection diffusion equation with inhomogeneous Dirichlet boundary is considered. By utilizing standard boundary shifting trick, a homogeneous boundary problem is derived with a singular source term which does not belong to \(L^2\) anymore. The variational formulation of such problem is established, based on which the finite element approximation scheme is developed. Inf-sup

In this paper we consider a mass- and energy–conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal \(L^{\infty }(H^1)\)-error estimates is still open, both in the

The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen–Cahn equation as a

The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method, the repeated solution to a shifted linear system of equations is necessary

This article establishes optimal upper and lower error estimates for strong full-discrete numerical approximations of the stochastic heat equation driven by space-time white noise. Thereby, this work proves the optimality of the strong convergence rates for certain full-discrete approximations of stochastic Allen–Cahn equations with space-time white noise which have been obtained in a recent previous

The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential \(\exp (A)v\) has to be approximated by \(\left( p_m(s^{-1}A)\right) ^sv=\exp (A+\varDelta A)v\). We propose here a rigorous bound for the relative backward error \(\left\Vert \varDelta A\right\Vert _{2}/\left\Vert

We consider a subdiffusion problem described by a time fractional Riemann–Liouville derivative of order \(0<\alpha <1\). The main purpose of this work is to show how we can apply fractional splines of order \(0<\beta \le 1\) to approximate a fractional integral and hence how to solve the subdiffusion problem using this approach. To discuss the convergence of the numerical method we present the error

The methods used for extracting an approximate eigenpair are crucial in sparse iterative eigensolvers. Using least squares and line search techniques this paper devises a method for an approximate eigenpair extraction. Numerical comparison of the Jacobi–Davidson method using the suggested method of eigenpair extraction, Rayleigh–Ritz, and refined Ritz projections shows that the suggested method is

In this paper we propose a dynamical low-rank strategy for the approximation of second order wave equations with random parameters. The governing equation is rewritten in Hamiltonian form and the approximate solution is expanded over a set of 2S dynamical symplectic-orthogonal deterministic basis functions with time-dependent stochastic coefficients. The reduced (low rank) dynamics is obtained by a

The semi-implicit Euler–Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change)

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented

In this paper, we study the numerical approximation of a coupled system of elliptic–parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the

An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications

In this paper a numerical multiscale method for discrete networks is presented. The method gives an accurate coarse scale representation of the full network by solving sub-network problems. The method is used to solve problems with highly varying connectivity or random network structure, showing optimal order convergence rates with respect to the mesh size of the coarse representation. Moreover, a

The eigenvector-dependent nonlinear eigenvalue problem arises in many important applications, such as the discretized Kohn–Sham equation in electronic structure calculations and the trace ratio problem in linear discriminant analysis. In this paper, we perform a perturbation analysis for the eigenvector-dependent nonlinear eigenvalue problem, which gives upper bounds for the distance between the solution

A formulation of the Taylor expansion with symmetric polynomial algebra allows to compute the coefficients of compact finite difference schemes, which solve the Poisson equation at an arbitrary order of accuracy on a uniform Cartesian grid in arbitrary dimensions. This construction produces original high order schemes which respect the Discrete Maximum Principle: a tenth order scheme in dimension three

The maximum angles \(\vartheta _q\) for which the three-, four-, five- and six-step backward difference formula methods are A(\(\vartheta _q\))-stable, slight improvements of the well-known angles, are determined.

Image resampling is a widely used tool in image processing. The upsampling increases the number of pixels and introduces new information to the image which can have undesired effects, like ringing artifacts and oscillations, aliasing “jagged” lines effect, or introduces too much numerical diffusion. Histopolation upsampling methods produce much sharper images but are more prone to aliasing and ringing

In this article, two radial basis functions based collocation schemes, differentiated and integrated methods (DRBF and IRBF), are extended to solve a class of nonlinear fractional initial and boundary value problems. Before discretization, the nonlinear problem is linearized using generalized quasilinearization. An interesting proof via generalized monotone quasilinearization for the existence and

We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to a Taylor series expansion approach previously used in the context of Nitsche’s method and, in the

In this study, a general formulation for the fractional-order general Lagrange scaling functions (FGLSFs) is introduced. These functions are employed for solving a class of fractional differential equations and a particular class of fractional delay differential equations. For this approach, we derive FGLSFs fractional integration and delay operational matrices. These operational matrices and collocation

Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro–macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages

A novel class of conservative numerical methods for general conservative Stratonovich stochastic differential equations with multiple invariants is proposed and analyzed. These methods, which are called modified averaged vector field methods, are constructed by modifying the averaged vector field method to preserve multiple invariants simultaneously. Based on the a prior estimate for high-order moments

The present article revisits the well-known stochastic theta methods (STMs) for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. Under a coupled monotonicity condition in a domain \(D \subset {{\mathbb {R}}}^d, d \in {{\mathbb {N}}}\), we propose a novel approach to achieve upper mean-square error bounds for STMs with the method parameters \(\theta

In this paper a special class of one-dimensional L-splines of order 4 is studied, which naturally appear in the computation of interpolation and smoothing with multivariate polysplines. Fast algorithms are provided for interpolation and smoothing with this class of L-splines, as well as a generalization of the Reinsch algorithm to this setting. The explicit description of all mathematical expressions

The article Jumping with variably scaled discontinuous kernels (VSDKs) written by S. De Marchi, F. Marchetti and E. Perracchione was originally published electronically on the publisher’s Internet portal on [date of OnlineFirst publication] without open access. With the author(s)’ decision to opt for Open Choice, the copyright of the article changed on January 2020 to © The Author(s) 2020 and the article

BDF formulas are among the most efficient methods for numerical integration, in particular of stiff equations (see e.g. Gear in Numerical initial value problems in ordinary differential equations, Prentice Hall, Upper Saddle River, 1971). Their excellent stability properties are known for precisely half a century, from the first calculation of their angles of \(A(\alpha )\)-stability by Nørsett (BIT

Analytic continuation is ill-posed, but becomes merely ill-conditioned (although with an infinite condition number) if it is known that the function in question is bounded in a given region of the complex plane. In an annulus, the Hadamard three-circles theorem implies that the ill-conditioning is not too severe, and we show how this explains the effectiveness of Chebfun and related numerical methods

A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix differential equation. A related algorithm is given for the approximation of symmetric or anti-symmetric time-dependent tensors by symmetric or anti-symmetric Tucker

Local Fourier analysis is a commonly used tool to assess the quality and aid in the construction of geometric multigrid methods for translationally invariant operators. In this paper we automate the process of local Fourier analysis and present a framework that can be applied to arbitrary, including non-orthogonal, repetitive structures. To this end we introduce the notion of crystal structures and

By piecewise Chebyshevian splines we mean splines with pieces taken from different Extended Chebyshev spaces all of the same dimension, and with connection matrices at the knots. Within this very large and crucial class of splines, we are more specifically concerned with those which are good for design, in the sense that they possess blossoms, or, equivalently, refinable B-spline bases. In practice

This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu (SIAM J Numer Anal 33:2417–2430, 1996). We provide a geometric Gauss–Newton method for solving the least squares inverse eigenvalue problem. The global

The post-processing of the solution of variational problems discretized with Galerkin finite element methods is particularly useful for the computation of quantities of interest. Such quantities are generally expressed as linear functionals of the solution and the error of their approximation is bounded by the error of the solution itself. Several a posteriori recovery procedures have been developed

Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie–Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate

We show that the minimizers of regularized quadratic functions restricted to their natural Krylov spaces increase in Euclidean norm as the spaces expand.

We derive bounds for the constants in Poincaré–Friedrichs inequalities with respect to mesh-dependent norms for complexes of discrete distributional differential forms. A key tool is a generalized flux reconstruction which is of independent interest. The results apply to piecewise polynomial de Rham sequences on bounded domains with mixed boundary conditions.

This paper presents the numerical analysis of a stabilized finite element scheme with discontinuous Galerkin (dG) discretization in time for the solution of a transient convection–diffusion–reaction equation in time-dependent domains. In particular, the local projection stabilization and the higher order dG time stepping scheme are used for convection dominated problems. Further, an arbitrary Lagrangian–Eulerian

In the originally published version, the author found some subsequent corrections. It should read correct.

In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy

The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard