Emergent behaviors of discrete Lohe aggregation flows

<p style='text-indent:20px;'>The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.</p>

A pseudo-spectral Strang splitting method for linear dispersive problems with transparent boundary conditions

The present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov–Galerkin method is considered which gives spectral accuracy. The main difficulty in constructing a second-order splitting scheme in such a situation lies in the compatibility condition at the boundaries of the sub-problems. In particular, the presence of an inflow boundary condition in the advection part results in order reduction. To overcome this issue a modified Strang splitting scheme is introduced that retains second-order accuracy. For this numerical scheme a stability analysis is conducted. In addition, numerical results are shown to support the theoretical derivations.

Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering

The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations.

A numerical investigation of Brockett’s ensemble optimal control problems

This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov–Tadmor method, a Runge–Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov–Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.

A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {\frac{k}{m}} for the convection part. With h the mesh width in space, this results in an error bound of the form {C_{0}h^{2}+C_{m}k} for appropriately smooth solutions, where {C_{m}\leq C^{\prime}+\frac{C^{\prime\prime}}{m}}. This work complements the earlier study [V. Thomée and A. S. Vasudeva Murthy, An explicit-implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 2019, 2, 283–293] based on the second-order Strang splitting.