A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

2007 ◽  
Vol 107 (1) ◽  
pp. 1-25 ◽  
Author(s):  
B. Bujanda ◽  
C. Clavero ◽  
J. L. Gracia ◽  
J. C. Jorge
Keyword(s):  
Reaction Diffusion ◽  
High Order ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Singularly Perturbed Problems ◽  
Alternating Direction ◽  
Uniformly Convergent

scholarly journals High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

2021 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Multistep Methods ◽  
General Setting ◽  
Reaction Diffusion ◽  
Strong Stability ◽  
High Order ◽  
Iterative Solvers ◽  
Time Dependent ◽  
Linear Multistep Methods ◽  
Advection Diffusion Equation ◽  
Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

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