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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?

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.

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

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.

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

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

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..