operator splitting
Many real PDEs bundle several different physical effects into one equation — advection plus diffusion plus reaction, say. Each effect on its own has a well-understood, efficient solver, but the combination is awkward to discretize all at once. Operator splitting is the pragmatic strategy of solving each piece in turn over a short time step, as if the others were briefly frozen, then cycling through them. You break the hard combined problem into a sequence of easy sub-problems you already know how to solve well.
Write the equation as u_t = (A + B) u, where A and B are the operators for two distinct effects (e.g. A = advection, B = diffusion). The simplest scheme, Lie splitting, advances over a step dt by first solving u_t = A u for the full step (using only the advection solver), then taking that result and solving u_t = B u for the full step (using only the diffusion solver). Doing A-then-B introduces a splitting error proportional to dt, because the two operators do not commute (advecting-then-diffusing is not the same as diffusing-then-advecting). The standard fix is Strang splitting, a symmetric sandwich: do a HALF step of A, then a FULL step of B, then a HALF step of A. The symmetry cancels the leading error term and the method becomes second-order accurate in time — a cheap and elegant upgrade.
Splitting is everywhere in large-scale simulation because it lets you assemble a complex solver from specialized, optimized building blocks — a Riemann solver for the advection part, an implicit solver for the stiff diffusion or reaction part, each running at the step size that suits it. It also tames stiffness: the stiff reaction term can be integrated implicitly while the non-stiff transport is done explicitly. The honest caveats: there is always a splitting error from non-commuting operators (so dt must be modest, and Lie splitting is only first order), boundary conditions can be subtle to split correctly, and for strongly coupled effects (where the operators interact intimately every instant) splitting can be inaccurate and a fully-coupled scheme is better.
For a reaction–diffusion equation u_t = D u_xx + R(u), Strang splitting over a step dt does: half a step of pure reaction (an ODE solver on u_t = R(u)), then a full step of pure diffusion (an implicit heat-equation solver), then another half step of reaction. Each sub-solver is best-in-class for its piece, yet the overall scheme is second-order in time.
Solve each physical effect with its own best solver, then alternate — symmetrically for second order.
Splitting error is unavoidable whenever the operators do not commute, so the answer depends on the step size and (for Lie splitting) the order you apply A and B. Strang splitting cancels the leading term but not all of it — splitting is not exact, only convergent.