Haijun Yu

Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

We develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by Smolyaks construction based on a nested quadrature. More precisely, we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis; and by using the aforementioned fast transform, we construct two very efficient sparse spectral-Galerkin methods for a model elliptic equation. In particular, the Chebyshev-Legendre-Galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low-order sparse grid methods based on finite elements or wavelets, and can be efficiently solved by a suitable sparse solver. Ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms.

Efficient energy stable numerical schemes for a phase field moving contact line model

In this paper, we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line. The model is a coupled system that consists of incompressible Navier-Stokes equations with a generalized Navier boundary condition and Cahn-Hilliard equation in conserved form. In both schemes the projection method is used to deal with the Navier-Stokes equations and stabilization approach is used for the non-convex Ginzburg-Landau bulk potential. By some subtle explicit-implicit treatments, we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions. An efficient spectral-Galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes. Numerical results show that the proposed schemes are very efficient and accurate. We proposed two linear energy stable numerical schemes for a phase field moving contact line model.Discrete energy laws that are consistent to the continuous ones are proved.The efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified.

Numerical approximations for a phase-field moving contact line model with variable densities and viscosities

We consider the numerical approximations of a two-phase hydrodynamics coupled phase-field model that incorporates the variable densities, viscosities and moving contact line boundary conditions. The model is a nonlinear, coupled system that consists of incompressible NavierStokes equations with the generalized Navier boundary condition, and the CahnHilliard equations with moving contact line boundary conditions. By some subtle explicitimplicit treatments to nonlinear terms, we develop two efficient, unconditionally energy stable numerical schemes, in particular, a linear decoupled energy stable scheme for the system with static contact line condition, and a nonlinear energy stable scheme for the system with dynamic contact line condition. An efficient spectral-Galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes. Various numerical results show that the proposed schemes are efficient and accurate.

Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Equations II. Unbounded Domains

This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [J. Shen and H. Yu, SIAM J. Sci. Comp., 32 (2010), pp. 3228-3250] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains and apply the new method to solve the electronic Schrodinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyaks algorithms based on nested, spectrally accurate quadratures and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients and allow us to construct a matrix-vector product algorithm with quasi-optimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms.

A New Spectral Element Method for Pricing European Options Under the Black-Scholes and Merton Jump Diffusion Models

We present a new spectral element method for solving partial integro-differential equations for pricing European options under the Black–Scholes and Merton jump diffusion models. Our main contributions are: (i)xa0using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii)xa0using Laguerre functions for the approximations on the semi-infinite domain, to avoid the domain truncation; and (iii)xa0deriving a rigorous proof of stability for the time discretizations of European put options under both the Black–Scholes model and the Merton jump diffusion model. The new method is flexible for handling different boundary conditions and non-smooth initial conditions for various contingent claims. Numerical examples are presented to demonstrate the efficiency and accuracy of the new method.

A dynamic-solver-consistent minimum action method

This paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method (MAM) for a spatially extended system when employing the large deviation principle (LDP) to study the effects of small random perturbations. A dynamic solver is used to approximate the unperturbed system, and a minimum action method is used to approximate the LDP, which corresponds to solving an Euler-Lagrange equation related to but more complicated than the unperturbed system. We will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the LDP, based on which we propose to define both the dynamic solver and the MAM on the same approximation space for spatial discretization. The semi-discrete LDP can then be regarded as the exact LDP of the semi-discrete unperturbed system, which is a finite-dimensional ODE system. We achieve this methodology for the two-dimensional Navier-Stokes equations using a divergence-free approximation space. The method developed can be used to study the nonlinear instability of wall-bounded parallel shear flows, and be generalized straightforwardly to three-dimensional cases. Numerical experiments are presented.

Model the nonlinear instability of wall-bounded shear flows as a rare event: a study on two-dimensional Poiseuille flow

In this work, we study the nonlinear instability of two-dimensional (2D) wall-bounded shear flows from the large deviation point of view. The main idea is to consider the Navier–Stokes equations perturbed by small noise in force and then examine the noise-induced transitions between the two coexisting stable solutions due to the subcritical bifurcation. When the amplitude of the noise goes to zero, the Freidlin–Wentzell (F–W) theory of large deviations defines the most probable transition path in the phase space, which is the minimizer of the F–W action functional and characterizes the development of the nonlinear instability subject to small random perturbations. Based on such a transition path we can define a critical Reynolds number for the nonlinear instability in the probabilistic sense. Then the action-based stability theory is applied to study the 2D Poiseuille flow in a short channel.

Approximations by orthonormal mapped Chebyshev functions for higher-dimensional problems in unbounded domains

This paper is concerned with approximation properties of orthonormal mapped Chebyshev functions (OMCFs) in unbounded domains. Unlike the usual mapped Chebyshev functions which are associated with weighted Sobolev spaces, the OMCFs are associated with the usual (non-weighted) Sobolev spaces. This leads to particularly simple stiffness and mass matrices for higher-dimensional problems. The approximation results for both the usual tensor product space and hyperbolic cross space are established, with the latter particularly suitable for higher-dimensional problems.

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

We propose a new weighted weak formulation for the Fokker–Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier–Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.

Efficient Second Order Unconditionally Stable Schemes for a Phase Field Moving Contact Line Model Using an Invariant Energy Quadratization Approach

We consider the numerical approximations for a phase field model consisting of incompressible Navier--Stokes equations with a generalized Navier boundary condition, and the Cahn--Hilliard equation ...