arXiv subjects

Ping Lin

Publications and source records attributed to Ping Lin.

44 records · Page 3Linked to original sources

arXiv2015

Pricing Two-asset Options under Exponential Lévy Model Using a Finite Element Method

This article presents a finite element method (FEM) for a partial integro-differential equation (PIDE) to price two-asset options with underlying price processes modeled by an exponential Levy process. We provide a variational formulation in a weighted Sobolev space, and establish existence and uniqueness of the FEM-based solution. Then we discuss the localization of the infinite domain problem to a finite domain and analyze its error. We tackle the localized problem by an explicit-implicit time-discretization of the PIDE, where the space-discretization is done through a standard continuous finite element method. Error estimates are given for the fully discretized localized problem where two assets are assumed to have uncorrelated jumps. Numerical experiments for the polynomial option and a few other two-asset options shed light on good performance of our proposed method.

q-fin.CP↗

arXiv2014

A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects

In this paper, we develop a phase-field model for binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. The governing equations of the model including the Navier-Stokes equations with additional stress term, Cahn-Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, some numerical examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared to the corresponding analytical solutions and the existing numerical results as validations for our model.

physics.flu-dyn↗

arXiv2014

A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law

In this paper, we investigate numerically a diffuse interface model for the Navier-Stokes equation with fluid-fluid interface when the fluids have different densities \cite{Lowengrub1998}. Under minor reformulation of the system, we show that there is a continuous energy law underlying the system, assuming that all variables have reasonable regularities. It is shown in the literature that an energy law preserving method will perform better for multiphase problems. Thus for the reformulated system, we design a $C^0$ finite element method and a special temporal scheme where the energy law is preserved at the discrete level. Such a discrete energy law (almost the same as the continuous energy law) for this variable density two-phase flow model has never been established before with $C^0$ finite element. A Newton's method is introduced to linearise the highly non-linear system of our discretization scheme. Some numerical experiments are carried out using the adaptive mesh to investigate the scenario of coalescing and rising drops with differing density ratio. The snapshots for the evolution of the interface together with the adaptive mesh at different times are presented to show that the evolution, including the break-up/pinch-off of the drop, can be handled smoothly by our numerical scheme. The discrete energy functional for the system is examined to show that the energy law at the discrete level is preserved by our scheme.

math-ph↗

arXiv2012

A priori and a posteriori $W^{1,\infty}$ error analysis of a QC method for complex lattices

In this paper we prove a priori and a posteriori error estimates for a multiscale numerical method for computing equilibria of multilattices under an external force. The error estimates are derived in a $W^{1,\infty}$ norm in one space dimension. One of the features of our analysis is that we establish an equivalent way of formulating the coarse-grained problem which greatly simplifies derivation of the error bounds (both, a priori and a posteriori). We illustrate our error estimates with numerical experiments.

math.NA↗

arXiv2012

Numerical Methods for Multilattices

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method has attracted growing interest in recent years. The QC method was first developed for crystalline materials with Bravais lattice and was later extended to multilattices (Tadmor et al, 1999). Another existing numerical approach to modeling multilattices is homogenization. In the present paper we review the existing numerical methods for multilattices and propose another concurrent macro-to-micro method in the numerical homogenization framework. We give a unified mathematical formulation of the new and the existing methods and show their equivalence. We then consider extensions of the proposed method to time-dependent problems and to random materials.

math.NA↗

arXiv2012

Quenching Time Optimal Control for Some Ordinary Differential Equations

This paper concerns some time optimal control problems of three different ordinary differential equations in $\mathbb{R}^2$. Corresponding to certain initial data and controls, the solutions of the systems quench at finite time. The goal to control the systems is to minimize the quenching time. To our best knowledge, the study on quenching time optimal control problems has not been touched upon. The purpose of this study is to obtain the existence and the Pontryagin maximum principles of optimal controls. We hope that our methods could hint people to study the same problems with more general vector fields in $\mathbb{R}^d$ with $d\in \mathbb{N}$. We also wish that our results could be extended to the same issue for parabolic equations.

math.OC↗

arXiv2010

Homogenization-based Analysis of Quasicontinuum Method for Complex Crystals

Among the efficient numerical methods based on atomistic models, the quasicontinuum (QC) method, introduced by Tadmor, Ortiz, and Phillips (1996), has attracted growing interest in recent years. Originally, the QC method was developed for materials with simple crystalline lattice (simple crystals) and later was extended to complex lattice (Tadmor et al, 1999). In the present paper we formulate the QC method for complex lattices in a homogenization framework and perform analysis of such a method in a 1D setting. We also present numerical examples showing that the convergence results are valid in a more general setting.

math.NA↗

arXiv2009

Energy-based ghost force removing techniques for the quasicontinuum method

This paper studies numerical methods for accurate treatment of the interface between the local and the nonlocal region in a QC approximation of atomistic materials. Only the energy-based methods are considered. Particularly, a quasicontinuum projection (QCP) method based on the idea of finite elements is shown to be accurate and efficient for this problem. We analyse the QCP method and study its relation to the existing methods, such as the quasinonlocal quasicontinuum method and the geometrically consistent reconstruction-based method. The analysis and the results of numerical tests confirm that the projection-based QC method successfully removes the ghost force with the same computational cost as the other methods. In all computed examples the error of QCP is either the same or lower as the error of the other methods. The performance of these methods in treating interfaces of elements in the local region is also examined.

math.NA↗