LARGE TIME STEP GENERALIZATION OF RANDOM CHOICE FINITE DIFFERENCE SCHEME FOR HYPERBOLIC CONSERVATION LAWS

A natural generalization of random choice finite difference scheme of Harten and Lax for Courant number larger than 1 is obtained. We handle interactions between neighboring Riemann solvers by linear superposition of their conserved quantities. We show consistency of the scheme for arbitrarily large Courant numbers. For scalar problems the scheme is total variation diminishing.A brief discussion is given for entropy condition.

ON THE QUASI-RANDOM CHOICE METHOD FOR THE LIOUVILLE EQUATION OF GEOMETRICAL OPTICS WITH DISCONTINUOUS WAVE SPEED

We study the quasi-random choice method （QRCM） for the Liouville equation of ge- ometrical optics with discontinuous locM wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discon- tinuities, and measure-valued solutions. The so-called QRCM is a random choice method based on quasi-random sampling （a deterministic alternative to random sampling）. The method not only is viscosity-free but also provides faster convergence rate. Therefore, it is appealing for the prob!em under study which is indeed a Hamiltonian flow. Our analy- sis and computational results show that the QRCM 1） is almost first-order accurate even with the aforementioned discontinuities; 2） gives sharp resolutions for all discontinuities encountered in the problem; and 3） for measure-valued solutions, does not need the level set decomposition for finite difference/volume methods with numerical viscosities.

Price Competition in the Random Coefficient Attraction Choice Models with Linear Cost

We study the pricing game between competing retailers under various random coefficient attraction choice models.We characterize existence conditions and structure properties of the equilibrium.Moreover,we explore how the randomness and cost parameters affect the equilibrium prices and profits under multinomial logit(MNL),multiplicative competitive interaction(MCI)and linear attraction choice models.Specifically,with bounded randomness,for the MCI and linear attraction models,the randomness always reduces the retailer’s profit.However,for the MNL model,the effect of randomness depends on the product’s value gap.For high-end products(i.e.,whose value gap is higher than a threshold),the randomness reduces the equilibrium profit,and vice versa.The results suggest high-end retailers in MNL markets exert more effort in disclosing their exact product performance to consumers.We also reveal the effects of randomness on retailers’pricing decisions.These results help retailers in making product performance disclosure and pricing decisions.