Achieving nonvanishing stability regions with high-gain cheap control using H/sup /spl infin// techniques: the second order case

1999 ◽  
Keyword(s):  
High Gain ◽  
Second Order ◽  
Stability Regions ◽  
Cheap Control ◽  
Order Case

scholarly journals Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Zhifeng Weng ◽  
Langyang Huang ◽  
Rong Wu
Keyword(s):  
Numerical Approximation ◽  
Second Order ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Backward Differentiation Formula ◽  
Hilliard Equation ◽  
Order Case ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Fourier Spectral Methods

In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

scholarly journals On correctness of first and second order fast marching method

2011 ◽  
Vol 1 (2) ◽  
Author(s):  
Christoph Lass
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Unique Solution ◽  
Nonlinear Partial Differential Equations ◽  
Second Order ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Order Case ◽  
Upwind Discretization ◽  
Marching Method

AbstractIn this article we will discuss the Fast Marching Method which was introduced by James A. Sethian to solve some types of nonlinear partial differential equations efficiently. We will show that this method yields the unique solution to an upwind discretization. Furthermore we will present the correct algorithm for the second order case where existence and unicity of the solution will be proven as well.

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