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We prove linear convergence for a new family of modified Dirichlet–Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin–Robin method. Such nonoverlapping domain decomposition methods are commonly employed for the parallelization of partial differential equation solvers. Convergence has been extensively studied for elliptic equations, but in the case of parabolic equations there are hardly any convergence results that are not relying on strong regularity assumptions. Hence, we construct a new framework for analyzing domain decomposition methods applied to quasilinear parabolic problems, based on fractional time derivatives and time-dependent Steklov–Poincaré operators. The convergence analysis is conducted without assuming restrictive regularity assumptions on the solutions or the numerical iterates. We also prove that these continuous convergence results extend to the discrete case obtained when combining domain decompositions with space-time finite elements. [ABSTRACT FROM AUTHOR]

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error E k ∞ : = (E sup j ∈ { 0 , … , N k } ‖ U (t j) - U j ‖ X p ) 1 / p → 0 (k → 0) , where p ∈ [ 2 , ∞) , U is the mild solution, U j is obtained from a time discretisation scheme, k is the step size, and N k = T / k for final time T > 0 . This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error E k : = (sup j ∈ { 0 , … , N k } E ‖ U (t j) - U j ‖ X p ) 1 / p , which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable. [ABSTRACT FROM AUTHOR]

This article deals with the nonlocal problem to a class of nonlinear non-autonomous second order integro-differential evolution equation of mixed type via measure of noncompactness in infinite-dimensional Banach spaces. Based on the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness combined with the theory of evolution families to investigate the existence of mild solutions for a class of nonlinear non-autonomous second order integro-differential evolution equations with nonlocal condition in infinite-dimensional Banach spaces, we obtained the existence of mild solutions under the weak situation that the nonlinear function satisfy some appropriate growth condition and non-compactness measure condition. Our results generalize and improve some previous results on this topic, since the condition of uniformly continuity of the nonlinearity is not required, and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted. Finally, an example is given to show the applications of the obtained results. [ABSTRACT FROM AUTHOR]

The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps. [ABSTRACT FROM AUTHOR]

Using the projection-difference method, we construct an approximate solution of an abstract linear parabolic equation in a separable Hilbert space with a periodic condition for a solution. We use the Galerkin method for the spatial variables and the implicit Euler discretization for time. We obtain root mean square estimates of the error of approximate solutions that are effective both in time and spatial variables; these estimates imply the convergence of approximate solutions to an exact solution and allow one to find the convergence rate. [ABSTRACT FROM AUTHOR]

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order α ∈ (0, 1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented. [ABSTRACT FROM AUTHOR]

Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than π 2 (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the L p - L r -smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. [ABSTRACT FROM AUTHOR]

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate these results with some numerical experiments. [ABSTRACT FROM AUTHOR]

We analyze a dimension splitting scheme for the time integration of linear Maxwell equations in a heterogeneous cuboid. The domain contains several homogeneous subcuboids and serves as a model for a rectangular embedded waveguide. Due to discontinuities of the material parameters and irregular initial data, the solution of the Maxwell system has regularity below H 1 . The splitting scheme is adapted to the arising singularities and is shown to converge with order one in L 2 . The error result only imposes assumptions on the model parameters and the initial data, but not on the unknown solution. To achieve this result, the regularity of the Maxwell system is analyzed in detail, giving rise to sharp explicit regularity statements. In particular, the regularity parameters are given in explicit terms of the largest jump of the material parameters. The analysis is based on semigroup theory, interpolation theory, and regularity analysis for elliptic transmission problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a discrete subdiffusion equation with memory. Based on the backward operator and the theory of fractional resolvent families, we find a discrete fractional resolvent sequence which allows to write the solution to this discrete subdiffusion equation as a variation of constant formula. [ABSTRACT FROM AUTHOR]

We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show an exponential convergence of the ODE solution to its steady state using a simple energy argument. At the discrete level, assuming the aforementioned step size restriction, the existence of an invariant set is proved and a matching exponential rate of convergence of the PAGD scheme is derived by mimicking the energy argument and the convergence at the continuous level. Applications of the PAGD method to numerical PDEs are demonstrated with certain nonlinear elliptic PDEs using pseudo-spectral methods for spatial discretization, and several numerical experiments are conducted. The results confirm the global geometric and mesh size-independent convergence of the PAGD method, with an accelerated rate that is improved over the preconditioned gradient descent (PGD) method. [ABSTRACT FROM AUTHOR]

This article analyses the convergence of the Lie–Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1 2 - in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie–Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities. [ABSTRACT FROM AUTHOR]

We construct and analyze a second-order implicit–explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate. [ABSTRACT FROM AUTHOR]

We consider the problem of waves propagating in a viscoelastic solid. For the material properties of the solid we consider both classical and fractional differentiation in time versions of the Zener, Maxwell, and Voigt models, where the coupling of different models within the same solid are covered as well. Stability of each model is investigated in the Laplace domain, and these are then translated to time-domain estimates. With the use of semigroup theory, some time-domain results are also given which avoid using the Laplace transform and give sharper estimates. We take the time to develop and explain the theory necessary to understand the relation between the equations we solve in the Laplace domain and those in the time-domain which are written using the language of causal tempered distributions. Finally we offer some numerical experiments that highlight some of the differences between the models and how different parameters effect the results.

In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the l ∞ (0 , T ; H) -stability of the BDF2 method for linear and semilinear parabolic equations is identical with the upper bound for the zero-stability. The l ∞ (0 , T ; V) -stability of the variable step-size BDF2 method is also established under more relaxed condition on the ratios of consecutive step-sizes. Based on these stability results, error estimates in several different norms are derived. To utilize the BDF method, the trapezoidal method and the backward Euler scheme are employed to compute the starting value. For the latter choice, order reduction phenomenon of the constant step-size BDF2 method is observed theoretically and numerically in several norms. Numerical results also illustrate the effectiveness of the proposed method for linear and semilinear parabolic equations. [ABSTRACT FROM AUTHOR]

In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions. [ABSTRACT FROM AUTHOR]

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schrödinger equation are included. [ABSTRACT FROM AUTHOR]

We present a framework for the systematic numerical approximation of nonlinear evolution problems with dissipation. The approach is based on rewriting the problem in a canonical form that complies with the underlying energy-dissipation structure. We show that the corresponding weak formulation then allows for a dissipation-preserving approximation by Galerkin methods in space and discontinuous Galerkin methods in time. The proposed methodology is rather general and can be applied to a wide range of applications. This is demonstrated by discussion of some typical examples ranging from diffusive partial differential equations to dissipative Hamiltonian systems. [ABSTRACT FROM AUTHOR]

We study the convergence of an inexact version of the classical Krasnosel'skii–Mann iteration for computing fixed points of nonexpansive maps. Our main result establishes a new metric bound for the fixed-point residuals, from which we derive their rate of convergence as well as the convergence of the iterates towards a fixed point. The results are applied to three variants of the basic iteration: infeasible iterations with approximate projections, the Ishikawa iteration, and diagonal Krasnosels'kii–Mann schemes. The results are also extended to continuous time in order to study the asymptotics of nonautonomous evolution equations governed by nonexpansive operators. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for an abstract quasilinear stochastic parabolic evolution equation on a Banach space driven by a cylindrical Brownian motion. We prove existence and uniqueness of a local strong solution up to a maximal stopping time, that is characterized by a blow-up alternative. The key idea is an iterative application of the theory about maximal Lp- regularity for semilinear stochastic evolution equations by Van Neerven, Veraar and Weis. We apply our local well-posedness result to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions and to a generalized Navier-Stokes equation describing non-Newtonian fluids. In the first example, we can even show that the solution exists globally. [ABSTRACT FROM AUTHOR]

Domain decomposition based time integrators allow the usage of parallel and distributed hardware, making them well-suited for the temporal discretization of parabolic systems. In this study, a rigours convergence analysis is given for such integrators without assuming any restrictive regularity on the solutions or the domains. The analysis is conducted by first deriving a new variational framework for the domain decomposition, which is applicable to the two standard degenerate examples. That is, the p-Laplace and the porous medium type vector fields. Secondly, the decomposed vector fields are restricted to the underlying pivot space and the time integration of the parabolic problem can then be interpreted as an operators splitting applied to a dissipative evolution equation. The convergence results then follow by employing elements of the approximation theory for nonlinear semigroups. [ABSTRACT FROM AUTHOR]

We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations. [ABSTRACT FROM AUTHOR]

This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated. [ABSTRACT FROM AUTHOR]

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result. [ABSTRACT FROM AUTHOR]

We discuss the existence and regularity of solutions to some fractional evolution equations in the q-norm. The linear part generates a noncompact semigroup, and the nonlinear part satisfies some conditions with respect to the fractional power norm of the linear part. In the end, we apply the obtained results to a control system. [ABSTRACT FROM AUTHOR]

The solvability of the abstract Cauchy problem for the quasilinear evolution equation u ′ ( t ) = A ( u ( t ) ) u ( t ) for t > 0 and u ( 0 ) = u 0 ∈ D is discussed. Here { A ( w ) ; w ∈ Y } is a family of closed linear operators in a real Banach space X such that Y ⊂ D ( A ( w ) ) ⊂ Y ¯ for w ∈ Y , Y is another Banach space which is continuously embedded in X , and D is a closed subset of Y . The existence and uniqueness of C 1 solutions to the Cauchy problem is proved without assuming that Y is dense in X or D ( A ( w ) ) is independent of w . The abstract result is applied to obtain an L 1 -valued C 1 -solution to a size-structured population model. [ABSTRACT FROM AUTHOR]

Preconditioned proximal penalty-duality two- and three-field algorithms for mixed optimality conditions, of evolution mixed constrained optimal control problems, are considered. Fixed point existence analysis is performed for corresponding evolution mixed governing variational state systems, in reflexive Banach spaces. Further, convergence analysis of the proximal penalty-duality algorithms is established via fixed point characterizations. In both analysis, a resolvent fixed point variational strategy is applied. [ABSTRACT FROM AUTHOR]

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed. [ABSTRACT FROM AUTHOR]

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates. [ABSTRACT FROM PUBLISHER]

We study semilinear evolution equations $$ \frac{\mathrm dU}{\mathrm dt}=AU+B(U)$$ posed on a Hilbert space $$\mathcal Y$$ , where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from $$\mathcal Y_\ell = D(A^\ell )$$ to itself, and $$\ell \in I \subseteq [0,L]$$ , $$L \ge 0$$ , $$0,L \in I$$ . In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge-Kutta method in time, retaining continuous space, and prove convergence of order $$O(h^{p\ell /(p+1)})$$ for non-smooth initial data $$U^0\in \mathcal Y_\ell $$ , where $$\ell \le p+1$$ , for a method of classical order p, extending a result by Brenner and Thomée for linear systems. Our approach is to project the semiflow and numerical method to spectral Galerkin approximations, and to balance the projection error with the error of the time discretization of the projected system. Numerical experiments suggest that our estimates are sharp. [ABSTRACT FROM AUTHOR]

In this article, we consider abstract linear conservative systems and their time-discrete counterparts. Our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time-discrete one. As an application, we show how observability properties for the time continuous model yield uniform (with respect to the time-step) observability results for its time-discrete approximation counterparts, provided the initial data are suitably filtered. The main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time-discrete models. Besides, using a reverse representation formula, we also prove that this estimate on the time of uniform observability for the time-discrete models is sharp. We then conclude with some general comments and open problems. [ABSTRACT FROM AUTHOR]

This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. The selection strategy consists of the following two stages. At the first stage, we employ explicit calculations for selecting the appropriate time step. At the second stage, using the implicit scheme, we calculate the solution at a new time level. This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side. The same formulas for the evaluation of the time step are obtained by using a comparison of two approximate solutions, which are obtained using the explicit scheme with the primary time step and the step that is reduced by half. Numerical results are presented for a model parabolic boundary value problem, which demonstrate the robustness of the developed algorithm for the time step selection. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the convergence analysis of the horizontal method of lines for evolution equations of the parabolic type. Following a semidiscretization in time by $$S$$ -stage one-step methods, the resulting elliptic stage equations per time step are solved with adaptive space discretization schemes. We investigate how the tolerances in each time step must be tuned in order to preserve the asymptotic temporal convergence order of the time stepping also in the presence of spatial discretization errors. In particular, we discuss the case of linearly implicit time integrators and adaptive wavelet discretizations in space. Using concepts from regularity theory for partial differential equations and from nonlinear approximation theory, we determine an upper bound for the degrees of freedom for the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance. [ABSTRACT FROM AUTHOR]

A convergence analysis is presented for the implicit Euler and Lie splitting schemes when applied to nonlinear parabolic equations with delay. More precisely, we consider a vector field which is the sum of an unbounded dissipative operator and a delay term, where both point delays and distributed delays fit into the framework. Such equations are frequently encountered, e.g. in population dynamics. The main theoretical result is that both schemes converge with an order (of at least) $$q=1/2$$ , without any artificial regularity assumptions. We discuss implementation details for the methods, and the convergence results are verified by numerical experiments demonstrating both the correct order, as well as the efficiency gain of Lie splitting as compared to the implicit Euler scheme. [ABSTRACT FROM AUTHOR]

We study convergence of the numerical methods in which the second order difference type method is combined with order two convolution quadrature for approximating the integral term of the evolutionary integral equation which arises in the theory of linear viscoelasticity. Here A is a positive self-adjoint densely defined linear operator in a real Hilbert space H and $\beta (t)$ is completely monotonic and locally integrable, but not constant. We establish the convergence properties of the discretization in time in the $l_{t}^{1}(0,\infty ;\,H)$ or $l_{t}^{\infty }(0,\infty ;\,H)$ norm. [ABSTRACT FROM AUTHOR]

We study the time discretization of the Cauchy problem where $$L$$ is a self-adjoint densely defined linear operator on a Hilbert space H with a complete eigen system $$\{\lambda _{m},\; \varphi _{m}\}_{m=1}^{\infty }$$, and the subscript denotes differentiation with respect to $$t$$. The real valued kernel $$\beta \in \,C(0,\,\infty )\bigcap \,L^{1}(0,\,1)$$ is assumed to be nonnegative, nonincreasing and convex, and $$-\beta ^{\prime }$$ is convex. The equation is discretized in time by Crank-Nicolson method based on the trapezoidal rule: while the time derivative is approximated by the trapezoidal rule in a two-step way, a convolution quadrature rule, constructed again from the trapezoidal rule, is used to approximate the integral term. The results and methods extend and simulate numerically those introduced by Carr and Hannsgen (SIAM J Math Anal 10:961-984, ) and (SIAM J Math Anal 13:459-483, ) for integrability with respect to continuous solutions. The uniform error estimates of the discretization in time are derived in the $$ l^{\infty }_{t}(0,\,\infty ;H) $$ norm. Some simple numerical examples illustrate our theoretical error bounds. [ABSTRACT FROM AUTHOR]

Let $${(r_{n})_{n \in \mathbb{N}}}$$ ( r n ) n ∈ N be the sequence of subdiagonal Padé approximations of the exponential function. We prove that for − A the generator of a uniformly bounded C0-semigroup T on a Banach space X, the sequence $${(r_{n}(-t A))_{n \in \mathbb{N}}}$$ ( r n ( - t A ) ) n ∈ N converges strongly to T( t) on D( Aα) for $${\alpha>\frac{1}{2}}$$ α > 1 2 . Local uniform convergence in t and explicit convergence rates in n are established. For specific classes of semigroups, such as bounded analytic or exponentially γ -stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given. [ABSTRACT FROM AUTHOR]

In this article we prove pathwise Hölder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem $$\begin{aligned} \left\{ \begin{aligned} dU(t)&= AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\ U(0)&=x_0. \end{aligned}\right. \end{aligned}$$ Here $$A$$ is the generator of an analytic $$C_0$$ -semigroup on a umd Banach space $$X,\,W_H$$ is a cylindrical Brownian motion in a Hilbert space $$H$$ , and the functions $$F:[0,T]\times X\rightarrow X_{\theta _F}$$ and $$G:[0,T]\times X\rightarrow {\fancyscript{L}}(H,X_{\theta _G})$$ satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise. [ABSTRACT FROM AUTHOR]