Spectral Properties of Effective Dynamics from Conditional Expectations

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.     

Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions

We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher’s equations—as well as their fractional generalizations—from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations.     

Hierarchy of equations to calculate the linear spectra of molecular aggregates: Time‐dependent and frequency domain formulation

In a recent publication (Ritschel et al., J. Chem. Phys. 2015, 142, 034115) we have derived a hierarchy of coupled differential equations in time domain to calculate the linear optical properties of molecular aggregates. Here, we provide details about issues concerning the numerical implementation. In addition we present the corresponding hierarchy in frequency domain.     

Singularity-free augmented Lagrangian algorithms for constrained multibody dynamics

After a general review of the methods currently available for the dynamics of constrained multibody systems in the context of numerical efficiency and ability to solve the differential equations of motion in singular positions, we examine the acceleration based augmented Lagrangian formulations, and propose a new one for holonomic and non-holonomic systems that is based on the canonical equations of Hamilton. This new one proves to be more stable and accurate that the acceleration based counterpart under repetitive singular positions. The proposed algorithms are numerically efficient, can use standard conditionally stable numerical integrators and do not fail in singular positions, as the classical formulations do. The reason for the numerical efficiency and better behavior under singularities relies on the fact that the leading matrix of the resultant system of ODEs is sparse, symmetric, positive definite, and its rank is independent of that of the Jacobian of the constraint equations. The latter fact makes the proposed method particularly suitable for singular configurations.     

横观各向同性饱和地基上中厚圆板的非轴对称振动

研究横观各向同性饱和土地基上中厚弹性圆板的非轴对称振动问题。基于横观各向同性饱和介质Biot波动方程的一般解,按混合边值问题建立了饱和地基与弹性中厚圆板非轴对称动力相互作用的对偶积分方程,并将对偶积分方程转化为易于计算的第二类Fredholm积分方程;采用数值方法求解该积分方程。数值算例结果表明,当h/a>0.05时,饱和半空间体上中厚度圆板在不同频率下的振动特性与相应频率下的刚性板的振动特性基本相同,当h/a<0.05时,板中心的位移将随h/a的减小而增大。     

Shocks in local supersonic zones

The results of numerical integration of the Euler equations governing two-dimensional and axisymmetric flows of an ideal (inviscid and non-heat-conducting) gas with local supersonic zones are presented. The subject of the study is the formation of shocks closing local supersonic zones. The flow in the vicinity of the initial point of the closing shock is calculated on embedded, successively refined grids with an accuracy much greater than that previously achieved. The calculations performed, together with the analysis of certain controversial issues, leave no doubt that it is the intersection of C ^?-characteristics proceeding from the sonic line inside the supersonic zone that is responsible for the closing shock formation.     

Efficient implicit algorithm for the equations of 2-D viscous compressible flow: application to shock-boundary layer interaction

A fully implicit algorithm has been developed to time integrate the equations of 2-D compressible viscous flow. The algorithm was constructed so as to optimize computational efficiency. The time-consuming block matrix inversions usually associated with implicit algorithms have been reduced to the trivial non-iterative inversion of four sets of scalar bidiagonal matrices. Thus, the algorithm requires virtually no more computer storage than an explicit algorithm. The efficient structure of the implicit algorithm is reflected in comparative timings which slow that it requires only a factor of two more computer time per point per time step than a typical explicit algorithm. Therefore, the algorithm allows more economical solution of given flows than existing explicit methods and also allows more difficult problems to be attempted using available computer resources. Application of the algorithm to the problem of shock-boundary layer interaction produces results consistent with both experimental measurements and other calculations.     

Doubly nonlinear parabolic-type equations as dynamical systems

In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some weak nonlinearities, also some mild nonlinearities of porous media type are allowed inside the time derivative. In order to formulate the equations as dynamical systems, some existence and uniqueness results are proved. Then the existence of a compact attractor is shown for a class of nonlinear PDEs that include doubly nonlinear porous medium-type equations. Under stronger smoothness assumptions on the nonlinearities, the finiteness of the fractal dimension of the attractor is also obtained.     

Stability and Synchronization of a Ring of Identical Cells with Delayed Coupling

We consider a ring of identical elements with time delayed, nearest neighbour coupling. The individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. The linear stability of the trivial solution is completely analyzed and illustrated in the parameter space of the coupling strength and the coupling delay. Conditions for global stability of the trivial solution are also given. The bifurcation and stability of nontrivial synchronous solutions from the trivial solution is analyzed using a centre manifold construction.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.