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Proper modeling of complex systems requires innovative mathematical tools. In this sense, we sought to use deformed or fractal derivatives for studying the dynamics of systems, particularly those, such as granular gases, in which the…
For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
An energy conservative discontinuous Galerkin scheme for a generalised third order KdV type equation is designed. Based on the conservation principle, we propose techniques that allow for the derivation of optimal a priori bounds for the…
This study proposes a novel spatial discretization procedure for the compressible Euler equations which guarantees entropy conservation at a discrete level when an arbitrary equation of state is assumed. The proposed method, based on a…
This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution…
A method for constructing first integral preserving numerical schemes for time-dependent partial differential equations on non-uniform grids is presented. The method can be used with both finite difference and partition of unity approaches,…
We use the optimized trigonometric finite basis method to find energy eigenvalues and eigenfunctions of the time-independent Schrodinger equation with high accuracy. We apply this method to the quartic anharmonic oscillator and the harmonic…
We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show…
In this paper, we propose a conservative local discontinuous Galerkin method for one-dimensional nonlinear Schr\"odinger equation. By using special upwind-biased numerical fluxes, we establish the optimal rate of convergence $\mathcal…
In this letter we apply a method recently devised in \cite{aapla03} to find precise approximate solutions to a certain class of nonlinear differential equations. The analysis carried out in \cite{aapla03} is refined and results of much…
Numerical models of weather and climate critically depend on long-term stability of integrators for systems of hyperbolic conservation laws. While such stability is often obtained from (physical or numerical) dissipation terms, physical…
In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A…
We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…
In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schr\"odinger-type system, which includes Schr\"odinger-Helmholz system and Schr\"odinger-Poisson system. In our numerical scheme, we…
In this article, a numerical analysis of the asymptotic behavior of the discrete energy associated to a dissipative coupled wave system is conducted. The numerical approximation of the system is constructed using the P1 finite element…
We propose a new approach for solving systems of conservation laws that admit a variational formulation of the time-discretized form, and encompasses the p-system or the system of elastodynamics. The approach consists of using constrained…
By recursively solving the underlying Schr\" odinger equation, we set up an efficient systematic approach for deriving analytic expressions for discretized effective actions. With this we obtain discrete short-time propagators for both one…