Splitting Methods for Parabolic and Hyperbolic Partial Differential Equations

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We will be concerned with the numerical solution of the parabolic partial differential equation (math)and also the first order hyperbolic partial differential equation $$\frac{{\partial u}}{{\partial t}} = L\left( {u,r,t,{D^2}} \right)u$$ where L is a linear operator (not the same operator in both equations), $$u = {\left( {{u_1},{u_2},...{u_n}} \right)^T}D = \left( {\frac{\partial }{{\partial {x_1}}},\frac{\partial }{{\partial {x_2}}},...,\frac{\partial }{{\partial {x_S}}}} \right)r = {\left( {{x_1},{x_2},...,{x_S}} \right)^T}$$ and where the matrix coefficients in L may depend on u, r and t. We assume the problems are well posed.