scholarly journals New Resolution Strategy for Multi-scale Reaction Waves using Time Operator Splitting and Space Adaptive Multiresolution: Application to Human Ischemic Stroke

2011 ◽  
Vol 34 ◽  
pp. 277-290 ◽  
Author(s):  
Max Duarte ◽  
Marc Massot ◽  
Stéphane Descombes ◽  
Christian Tenaud ◽  
Thierry Dumont ◽  
...  
Keyword(s):  
Ischemic Stroke ◽  
Operator Splitting ◽  
Time Operator ◽  
Multi Scale ◽  
Resolution Strategy

scholarly journals New Resolution Strategies for Multi-scale Reaction Waves: Optimal Time Operator Splitting and Space Adaptive Multiresolution

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Max Duarte ◽  
Marc Massot ◽  
Frédérique Laurent ◽  
Stéphane Descombes ◽  
Christian Tenaud ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Source Term ◽  
Operator Splitting ◽  
Reaction Diffusion ◽  
Time Operator ◽  
Reaction Diffusion Equations ◽  
Optimal Time ◽  
Time Step ◽  
Multi Scale ◽  
Reaction Fronts

We tackle the numerical simulation of reaction-diffusion equations modeling multi-scale reac- tion waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reaction fronts, spatially very lo- calized. In this paper, we introduce a new resolution strategy based on time operator splitting and space adaptive multiresolution in the context of very localized and stiff reaction fronts. Based on recent theoretical studies of numerical analysis, such a strategy leads to a splitting time step which is not restricted neither by the fastest scales in the source term nor by restric- tive diffusive step stability limits, but only by the physics of the phenomenon. We thus aim at solving accurately complete models including all time and space scales of the phenomenon, considering large simulation domains with conventional computing resources. The efficiency is evaluated through the numerical simulation of configurations which were so far out of reach of standard methods in the field of nonlinear chemical dynamics for 2D spiral waves and 3D scroll waves as an illustration. Future extensions of the proposed strategy are finally discussed.

An efficient space-time operator-splitting method for high-dimensional vector-valued Allen–Cahn equations

2019 ◽  
Vol 29 (9) ◽  
pp. 3437-3453 ◽  
Author(s):  
Yunxia Sun ◽  
Xufeng Xiao ◽  
Zhiming Gao ◽  
Xinlong Feng
Keyword(s):  
Upper Bound ◽  
Operator Splitting ◽  
Splitting Method ◽  
Mass Conservation ◽  
Space Time ◽  
Time Operator ◽  
High Dimensional ◽  
Heat Equations ◽  
Content Type ◽  
Operator Splitting Method

Purpose The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations. Design/methodology/approach Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement. Findings The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method. Practical implications Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method. Originality/value The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

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