THE LIMITS OF APPLICABILITY OF THE CANONICAL OPERATOR METHOD FOR NONSTRICTLY HYPERBOLIC EQUATIONS WITH NONSMOOTH CHARACTERISTICS

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

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Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽

10.3390/math9131483 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1483

Author(s):

Shanqin Chen

Keyword(s):

Large Time ◽

Hyperbolic Equations ◽

Strong Stability ◽

Matrix Exponential ◽

Linear Component ◽

Nonlinear Hyperbolic Equations ◽

Time Step ◽

Integrating Factor ◽

The Matrix ◽

Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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A Dynamical Study of Fusion Hindrance with Nakajima-Zwanzig Projection Method

Progress of Theoretical and Experimental Physics ◽

10.1093/ptep/ptab005 ◽

2021 ◽

Author(s):

Yasuhisa Abe ◽

David Boilley ◽

Quentin Hourdillé ◽

Caiwan Shen

Keyword(s):

Cross Sections ◽

Degrees Of Freedom ◽

Operator Method ◽

Initial Distribution ◽

Physical Parameters ◽

Relaxation Processes ◽

Fast Variable ◽

Dynamical Study ◽

Injection Point ◽

Slow Variables

Abstract A new framework is proposed for the study of collisions between very heavy ions which lead to the synthesis of Super-Heavy Elements (SHE), to address the fusion hindrance phenomenon. The dynamics of the reaction is studied in terms of collective degrees of freedom undergoing relaxation processes with different time scales. The Nakajima-Zwanzig projection operator method is employed to eliminate fast variable and derive a dynamical equation for the reduced system with only slow variables. There, the time evolution operator is renormalised and an inhomogeneous term appears, which represents a propagation of the given initial distribution. The term results in a slip to the initial values of the slow variables. We expect that gives a dynamical origin of the so-called “injection point s” introduced by Swiatecki et al in order to reproduce absolute values of measured cross sections for SHE. A formula for the slip is given in terms of physical parameters of the system, which confirms the results recently obtained with a Langevin equation, and permits us to compare various incident channels.

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