Related papers: A-stable Runge-Kutta methods for semilinear evolut…

A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^\gamma X(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $\gamma \in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert…

A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge-Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach…

We discuss a semi-discrete analogue of the Unified Transform Method, introduced by A. S. Fokas, to solve initial-boundary-value problems for linear evolution partial differential equations of constant coefficients. The semi-discrete method…

We consider the geometric evolution problem of entire graphs moving by fractional mean curvature. For this, we study the associated nonlocal quasilinear evolution equation satisfied by the family of graph functions. We establish, using an…

Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the…

In this paper we prove the local and global well-posedness of the time fractional abstract Schr\"odinger type evolution equation on the Hilbert space and as an application, we prove the local and global well-posedness of the fractional…

We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes…

We deal with a class of semilinear nonlocal differential equations in Hilbert spaces which is a general model for some anomalous diffusion equations. By using the theory of integral equations with completely positive kernel together with…

In this paper, we obtain some stability results of (abstract) dissipative evolution equations with a nonautonomous and nonlinear damping using the exponential stability of the retrograde problem with a linear and autonomous feedback and a…

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension, and also the analogous problem for a symmetric variant of the system. Assuming smoothness of solutions, we discretize these problems…

Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of…

In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…

In this paper a new Runge-Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a…

This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first…

We consider the Shallow Water equations in the supercritical and subcritical cases in one space variable,posed in a finite spatial interval with characteristic boundary conditions at the endpoints, which, as is well known, are…

We perform a numerical free evolution of a selfgravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in Eddington-Finkelstein…

This paper deals with stability of classical Runge-Kutta collocation methods. When such methods are embedded in linearly implicit methods as developed in [12] and used in [13] for the time integration of nonlinear evolution PDEs, the…

We generalize our earlier results concerning meshfree collocation methods for semilinear elliptic second order problems to the quasilinear case. The stability question, however, is treated differently, namely by extending a paper on…

A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where…

In this paper, we propose Galerkin-Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrodinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a…