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The aim of this paper is to develop and analyze high-order time stepping schemes for solving semilinear subdiffusion equations. We apply the $k$-step BDF convolution quadrature to discretize the time-fractional derivative with order…
Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid…
Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the…
High-order numerical methods for solving elliptic equations over arbitrary domains typically require specialized machinery, such as high-quality conforming grids for finite elements method, and quadrature rules for boundary integral…
We are investigating the first strong convergence analysis of a numerical method for stochastic differential algebraic equations (SDAEs) under a non-global Lipschitz setting. It is well known that the explicit Euler scheme fails to converge…
Multirate behavior of ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) is characterized by widely separated time constants in different components of the solution or different additive terms of the…
A pure frequency domain method for the computation of periodic solutions of nonlinear ordinary differential equations (ODEs) is proposed in this study. The method is particularly suitable for the analysis of systems that feature distinct…
In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) scheme to adapt them to systems of ODEs. This leads to exact schemes in the linear case, and also improve the accuracy in the nonlinear…
We use the semi-discrete method, originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6), to reproduce qualitative…
Recently, Martin Hutzenthaler pointed out that the explicit Euler method fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with superlinearly growing and globally one sided Lipschitz drift…
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide…
In this paper implicit and explicit exact difference schemes (EDS) for system $\textbf{x}' = A\textbf{x}$ of three linear differential equations with constant coefficients are constructed. Numerical simulations for stiff problem and for…
Implicit methods for the numerical solution of initial-value problems may admit multiple solutions at any given time step. Accordingly, their nonlinear solvers may converge to any of these solutions. Below a critical timestep, exactly one…
This paper is the second in a series of works on weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a…
This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs…
When the singular values of the evolution operator are all smaller or all greater than one, stable integration algorithms are obtained either by explicit or implicit methods. When the singular spectrum mixes greater and smaller than one…
Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one-dimensional dynamic systems. In this paper we propose a smoothing approach regularized by a quasilinearized ODE-based penalty in…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…
Radau IIA methods, specifically the adaptive order Radau method in Fortran due to Hairer, are known to be state-of-the-art for the high-accuracy solution of highly stiff ordinary differential equations (ODEs). However, the traditional…