Higher Order Operator Splitting Method

In summary, the conversation revolves around the possibility of using exponential form for non-linear operator structures in the solution of deterministic parabolic equations. The link provided shows the exponential form of the solution and the inclusion of the non-linear operator structure in operator "D". The asker is seeking a mathematical demonstration to show that such exponential form can be used for non-linear operators. They mention that the exponential form is consistent for linear operators but not for non-linear operators, which raises confusion. They also express their disappointment for not receiving any responses to their question yet and ask for additional resources on the topic.
  • #1
mertcan
344
6
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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  • #2
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
 
  • #3
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 ...
 
  • #4
As it seems to me, I won't have responses but at least could you provide me with nice sources links files videos...?
 

1. What is the Higher Order Operator Splitting Method?

The Higher Order Operator Splitting Method is a numerical method used to solve partial differential equations (PDEs). It is a type of splitting method, meaning that it breaks down the PDE into smaller, simpler problems that can be solved separately.

2. How does the Higher Order Operator Splitting Method work?

The Higher Order Operator Splitting Method works by splitting the PDE into smaller problems, each of which can be solved using a different numerical method. The solutions from each smaller problem are then combined to approximate the solution to the original PDE.

3. What are the advantages of using the Higher Order Operator Splitting Method?

One advantage of the Higher Order Operator Splitting Method is that it allows for more accurate solutions compared to other splitting methods. It can also handle a wider range of PDEs, including non-linear and time-dependent equations.

4. What are the limitations of the Higher Order Operator Splitting Method?

One limitation of the Higher Order Operator Splitting Method is that it can be computationally expensive, as it requires solving multiple smaller problems. It may also lose accuracy when dealing with stiff problems, where the solution changes rapidly over a small time interval.

5. In what applications is the Higher Order Operator Splitting Method commonly used?

The Higher Order Operator Splitting Method is commonly used in applications such as fluid dynamics, chemical reactions, and heat transfer problems. It is also used in fields such as computational physics, engineering, and climate modeling.

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