Chiu-Yen Kao

Chiu-Yen Kao, Yuji Kodama

The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions in this paper. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. In this paper, we study the initial value problem of the KP equation with V- and X-shape initial waves consisting of two distinct line-solitons by means of the direct numerical simulation. We then show that the solution converges asymptotically to some of those exact soliton solutions. The convergence is in a locally defined $L^2$-sense. The initial wave patterns considered in this paper are related to the rogue waves generated by nonlinear wave interactions in shallow water wave problem.

Ying Wang, Chiu-Yen Kao

The focus of the present study is the modified Buckley-Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley-Leverett (BL) equation by including a balanced diffusive-dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that the solution of the finite interval [0,L] boundary value problem converges to that of the half-line [0,+\infty) boundary value problem for the MBL equation as L-> +\infty. This result provides a justification for the use of the finite interval boundary value problem in numerical studies for the half line problem. Furthermore, we extend the classical central schemes for the hyperbolic conservation laws to solve the MBL equation which is of pseudo-parabolic type. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.

Xiaobing Feng, Chiu-Yen Kao, Thomas Lewis

This paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations (PDEs) in 1-D. The goal of the paper is to extend the successful framework of monotone, consistent, and stable finite difference methods for first order fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs such as Monge-Ampère and Bellman type equations. New concepts of consistency, generalized monotonicity, and stability are introduced; among them, the generalized monotonicity and consistency, which are easier to verify in practice, are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear Hamilton-Jacobi equations. The main component of the proposed framework is the concept of "numerical operator", and the main idea used to design consistent, monotone and stable finite difference methods is the concept of "numerical moment". These two new concepts play the same roles as the "numerical Hamiltonian" and the "numerical viscosity" play in the finite difference framework for first order fully nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also are proved to be stable and the second class contains Godunov-like methods. Numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper.

Weaam Alhejaili, Chiu-Yen Kao

In this paper, a spectral method based on conformal mappings is proposed to solve Steklov eigenvalue problems and their related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use the gradient ascent approach to find the optimal domain which maximizes $k-$th Steklov eigenvalue with a fixed area for a given $k$. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different $k$