Given an evolution PDE$$u_t = Au + Bu$$where $A,B$ are (possibly nonlinear) differential operators that don't commute, a common numerical approach is to alternate between solving$$u_t = Au$$and$$u_t = Bu.$$The simplest implementation of this is known as Godunov splitting and is 1st-order accurate.  Another well-known approach, known as Strang splitting, is 2nd-order accurate.  Do higher order operator splitting methods (or alternative multiphysics discretization approaches) exist?
Are there operator splitting approaches for multiphysics PDEs that achieve high order convergence?
pde;multiphysics;operator splitting
It was my understanding that the BCH formula was a systematic way to approximate the matrix exponential of two non-commutative matrices.