This paper presents a new immersed finite volume element method for solving second-order elliptic problems with discontinuous diffusion coefficient on a Cartesian mesh. The new method possesses the local conservation property of classic finite volume element method, and it can overcome the oscillating behaviour of the classic immersed finite volume element method. The idea of this method is to reconstruct the control volume according to the interface, which makes it easy to implement. Optimal error estimates can be derived with respect to an energy norm under piecewise $H^{2}$ regularity. Numerical results show that the new method significantly outperforms the classic immersed finite volume element method, and has second-order convergence in $L^{\infty }$ norm.

The Kirchhoff approximation is widely used to describe the scatter of elastodynamic waves. It simulates the scattered field as the convolution of the free-space Green’s tensor with the geometrical elastodynamics approximation to the total field on the scatterer surface and, therefore, cannot be used to describe nongeometrical phenomena, such as head waves. The aim of this paper is to demonstrate that an alternative approximation, the convolution of the far-field asymptotics of the Lamb’s Green’s tensor with incident surface tractions, has no such limitation. This is done by simulating the scatter of a critical Gaussian beam of transverse motions from an infinite plane. The results are of interest in ultrasonic nondestructive testing.

We consider the numerical solution of competitive exothermic and endothermic reactions in the presence of a chaotic advection flow. The resulting behaviour is characterized by a strong dependence on the competitive reaction history. The burnt temperature is not immediately connected to simple enthalpy calculations, so there is a subtlety in the interplay between the major parameters, notably the Damköhler number, the ratio of the heats of exothermic and endothermic reactions, as well as the ratio of their respective activation energies. This paper seeks to explore the way these parameters affect the steady states of these reaction fronts and their stability.

In 2015, Guglielmi and Badia discussed optimal strategies in a particular type of service system with two strategic servers. In their setup, each server can be either active or inactive and an active server can be requested to transmit a sequence of packets. The servers have varying probabilities of successfully transmitting when they are active, and both servers receive a unit reward if the sequence of packets is transmitted successfully. Guglielmi and Badia provided an analysis of optimal strategies in four scenarios: where each server does not know the other’s successful transmission probability; one of the two servers is always inactive; each server knows the other’s successful transmission probability and they are willing to cooperate.

Unfortunately, the analysis by Guglielmi and Badia contained some errors. In this paper we correct these errors. We discuss three cases where both servers (I) communicate and cooperate; (II) neither communicate nor cooperate; (III) communicate but do not cooperate. In particular, we obtain the unique Nash equilibrium strategy in Case II through a Bayesian game formulation, and demonstrate that there is a region in the parameter space where there are multiple Nash equilibria in Case III. We also quantify the value of communication or cooperation by comparing the social welfare in the three cases, and propose possible regulations to make the Nash equilibrium strategy the socially optimal strategy for both Cases II and III.

We investigate convergence in the cone of completely monotone functions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials.

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.