  We establish a novel convergent iteration framework for a weak approximation
of general switching diffusion. The key theoretical basis of the proposed
approach is a restriction of the maximum number of switching so as to untangle
and compensate a challenging system of weakly coupled partial differential
equations to a collection of independent partial differential equations, for
which a variety of accurate and efficient numerical methods are available.
Upper and lower bounding functions for the solutions are constructed using the
iterative approximate solutions. We provide a rigorous convergence analysis for
the iterative approximate solutions, as well as for the upper and lower
bounding functions. Numerical results are provided to examine our theoretical
findings and the effectiveness of the proposed framework.
