Graduate Higher Order Operator Splitting Method

Click For Summary
The discussion revolves around the application of the exponential form of solutions in higher-order operator splitting methods, particularly for deterministic parabolic equations. The user seeks clarification on how the exponential form can be applied when the operator "D" includes non-linear structures, specifically when x is dependent on t. They note that while the first-order expansion aligns with expectations, the second-order expansion fails to maintain consistency due to the presence of non-linear operators. The user expresses frustration over the lack of responses and requests additional resources or explanations to better understand the applicability of the exponential form in non-linear contexts. The inquiry highlights a gap in understanding the mathematical foundations of operator splitting methods when non-linearities are involved.
mertcan
Messages
343
Reaction score
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##
 
Mathematics news on Phys.org
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...?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Undergrad Using Differential operators to solve Diff equations
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Method for finding a complex series?
  • · Replies 10 ·
Replies
10
Views
2K
High School Complex products: perpendicular vectors and rotation effects
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Second order non-homogeneous linear ordinary differential equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad The interpretation of probability
  • · Replies 42 ·
2
Replies
42
Views
7K
Undergrad Linear least-squares method and row multiplication of matrix
  • · Replies 2 ·
Replies
2
Views
2K
Graduate The eigenvalue power method for quantum problems
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Equality of two particular solutions of 2nd order linear ODE
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Eigenvalues for a non self adjoint operator
  • · Replies 4 ·
Replies
4
Views
3K
How Do Complex Numbers Extend the Real Number System?
  • · Replies 2 ·
Replies
2
Views
3K