What is the operator splitting technique?

Cuchulainn · March 1st, 2012, 2:20 pm

Fractional steps/splitting QuoteSo, here are two schemes for solving u_t=u_xx+u_yy:Leg1: solve 1/2u_t=u_xx i.e. (u^n+1-u^n)/0.5dt=L*u, L is a discretization matrix for u_xx initial data is givenLeg2: solve 1/2u_t=u_yy i.e. (u^n+1-u^n)/0.5dt=H*u, H is a discretization matrix for u_yy with initial data from Leg1.which technically means do HALF of the step one leg and another step another leg. assign to u^n+1 value from leg 2;Is this a kind of Strang scheme? It is not used in the Russian literature AFAIK. What's the truncation error of that scheme?

Tene · May 12th, 2012, 6:33 pm

The attached write up gives one way to show that Strang symmetrized operatorsplitting method is second-order accurate also for non constant coefficient casewithout assuming commuting operators.

Cuchulainn · May 13th, 2012, 5:32 pm

Does this method work with mixed derivatives. i.e. A = A1 + A2? What would A1 and A2 be for let's say Heston pde?

Tene · May 14th, 2012, 4:37 pm

Yes the method should work also for problems with a mixed derivative.Although computationally two-way splitting would not be convenientin this case.With a mixed derivative, I would do three-way splitting A = A1 + A2 + A3,where A1 would be the x-direction, A2 would be the y-direction, and A3would a diagonal direction. Performing a Strang symmetrization for thissplitting leads to 5 substeps. I believe that this three-way splitting methodwould be second-order accurate also with time-dependent coefficients.

jfuqua · October 26th, 2012, 1:09 pm

High order splitting schemes with complex timesteps and their application in mathematical financehttp://arxiv.org/abs/1210.5392

jfuqua · November 7th, 2012, 7:39 pm

This may be of interest/use.Chertock Alina, Alexander Kurganov 'On Splitting-Based Numerical Methods for Convection-Diffusion Equations' http://www4.ncsu.edu/~acherto/papers/Ch ... rganov.pdf

Cuchulainn · November 9th, 2012, 8:46 am

Thanks for the link. More attention to Strang-Marchuk splitting is good.In general, there is no a-priori reason IMO why Strang-Marchuk splitting should be able to resolve convection-dominance without special treatment. Here is SM splitting from the perspective of Lie Theory (5)-(7)Strang for BSMaybe (36) can be simplified..