Higher Order Operator Splitting Method

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Discussion Overview

The discussion centers on the application of higher-order operator splitting methods for deterministic parabolic equations, specifically focusing on the use of exponential forms of solutions when non-linear operators are involved. Participants are exploring the mathematical consistency of these methods in the context of both linear and non-linear operators.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the validity of using the exponential form of the solution when the operator "D" includes non-linear components, specifically mentioning the operator structure ##\partial_x x##.
  • The same participant notes that while the exponential form is consistent for first-order expansions, it becomes inconsistent for second-order expansions when non-linear operators are included.
  • Another participant expresses frustration over the lack of responses to their inquiry about the applicability of exponential forms to non-linear operators.
  • There is a request for additional resources or references that could help clarify the topic further.

Areas of Agreement / Disagreement

The discussion remains unresolved, with no consensus on whether the exponential form can be applied to non-linear operators. Participants express differing levels of understanding and seek clarification.

Contextual Notes

Participants have not reached an agreement on the mathematical demonstration needed to show the applicability of the exponential form to non-linear operators. There are also indications of missing assumptions regarding the dependence of variables on time.

Who May Find This Useful

Researchers and students interested in operator splitting methods, deterministic parabolic equations, and the mathematical foundations of non-linear operator theory may find this discussion relevant.

mertcan
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Hi, in the link https://www.researchgate.net/profile/Andrew_Sornborger/publication/220662120_Higher-order_operator_splitting_methods_for_deterministic_parabolic_equations/links/568ffaab08aec14fa557b85e/Higher-order-operator-splitting-methods-for-deterministic-parabolic-equations.pdf and equation 3 you will see the exponential form of the solution, also operator "D" includes non-linear operator structure written as ##\partial_x x##. Besides, I think in previous link x is dependent on t. I can understand that when operator "D" only consists of linear operators like ##\partial_x## then exponential form is consistent but I can not understand this situation is also possible when non-linear operators are included in "D". Could you provide me with mathematical demonstration to show that exponential form can be written also for non-linear operator structures??

When I expand exponential form of operator "D", I can see it is totally consistent with first order but when I come to second order expansion of operator "D" then it is not going well and not consistent contrary to link I shared. If "D" only included linear operators, it would be ok but here we have nonlinear operator, x depends on t so at the second order it can not be written as 1/2*D^2*##\Delta_t##
 
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Hi everyone it has been 3 days since I posted my that thread so still no response. please let me know if there is not understanble part in my question... I would be glad o provide additional infos
 
I do not know why I can not get responses but my question is so simple and humble: we can use exponential form of linear operators but can we use the same exponential form for NONLINEAR operators?? For more info can look at my post 1 ...
 
As it seems to me, I won't have responses but at least could you provide me with nice sources links files videos...?
 

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