Question

I am looking at equation u_t=a(t,x,y)u_xx+b(t,x,y)u_yy and I want to investigate the splitting error that occurs if I apply sequential splitting such as solve first u_t=a(t,x,y)u_xx and then u_t=b(t,x,y)u_yy with initial data from step 1. This method is globally first order due to splitting. Usually the splitting error is shown via semi group theory, as exponential and there difference. However, if I have variable coefficients I can't use the same tool. I don't know how to prove such a statement even what are my possible approaches? Please let me know if anybody has an idea. Thanks!