A Higher Order Operator Splitting Method

AI Thread 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 'Video on imaginary numbers and some queries'

Thread 'Video on imaginary numbers and some queries'

Hi, I was watching the following video. I found some points confusing. Could you please help me to understand the gaps? Thanks, in advance! Question 1: Around 4:22, the video says the following. So for those mathematicians, negative numbers didn't exist. You could subtract, that is find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »
Thread 'Unit Circle Double Angle Derivations'

Thread 'Unit Circle Double Angle Derivations'

Here I made a terrible mistake of assuming this to be an equilateral triangle and set 2sinx=1 => x=pi/6. Although this did derive the double angle formulas it also led into a terrible mess trying to find all the combinations of sides. I must have been tired and just assumed 6x=180 and 2sinx=1. By that time, I was so mindset that I nearly scolded a person for even saying 90-x. I wonder if this is a case of biased observation that seeks to dis credit me like Jesus of Nazareth since in reality...
View full post »

Similar threads

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

Hot Threads

Recent Insights

Back
Top