High-Order Approximation of Heteroclinic Bifurcations in Truncated 2D-Normal Forms for the Generic Cases of Hopf-Zero and Nonresonant Double Hopf Singularities

2021 ◽  
Vol 20 (1) ◽  
pp. 403-437
Author(s):  
B. W. Qin ◽  
K. W. Chung ◽  
A. Algaba ◽  
A. J. Rodríguez-Luis
Keyword(s):  
Normal Forms ◽  
Order Approximation ◽  
High Order ◽  
High Order Approximation

The SKN Approximation for Solving Radiative Transfer Problems in Absorbing, Emitting, and Isotropically Scattering Plane-Parallel Medium: Part 1

2002 ◽  
Vol 124 (4) ◽  
pp. 674-684 ◽  
Author(s):  
Zekeriya Altac¸
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Differential Equations ◽  
Order Approximation ◽  
High Order ◽  
High Order Approximation ◽  
One Dimensional ◽  
Second Order Differential Equations ◽  
Plane Parallel ◽  
Transfer Problems

A high order approximation, the SKN method—a mnemonic for synthetic kernel—is proposed for solving radiative transfer problems in participating medium. The method relies on approximating the integral transfer kernel by a sum of exponential kernels. The radiative integral equation is then reducible to a set of coupled second-order differential equations. The method is tested for one-dimensional plane-parallel participating medium. Three quadrature sets are proposed for the method, and the convergence of the method with the proposed sets is explored. The SKN solutions are compared with the exact, PN, and SN solutions. The SK1 and SK2 approximations using quadrature Set-2 possess the capability of solving radiative transfer problems in optically thin systems.

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