Related papers: Splitting methods for constrained diffusion-reacti…
This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, time-subcycling techniques are widely used every day to improve…
We consider nonlinear drift-diffusion equations (both porous medium equations and fast diffusion equations) with a measure-valued external force. We establish existence of nonnegative weak solutions satisfying gradient estimates, provided…
The Schr\"odinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into…
Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter (e.g., the temperature) that depends on integrals of the unknown…
We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resluting algorithm is a an extension of the algorithm in [4]. The…
We deal with various splitting methods in algebraic logic. The word `splitting' refers to splitting some of the atoms in a given relation or cylindric algebra each into one or more subatoms obtaining a bigger algebra, where the number of…
An initial-boundary value problem for the $n$-dimensional ($n\geq 2$) time-dependent Schr\"odinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we…
Abstract argumentation frameworks are formal systems that facilitate obtaining conclusions from non-monotonic knowledge systems. Within such a system, an argumentation semantics is defined as a set of arguments with some desired qualities,…
As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear…
We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
Convergence of the solutions of nonhomogeneous linear singularly perturbed systems to that of the corresponding reduced singular system on the half-line [0, $\infty $) is considered. To include the situation on a neighborhood of initial…
A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new…
Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces),…
Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion…
A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly…
We present a method for constructing numerical schemes with up to 3rd strong convergence order for solution of a class of stochastic differential equations, including equations of the Langevin type. The construction proceeds in two stages.…
A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the…