ABSTRACT
This chapter addresses space-time coupled methods as well as spacetime decoupled methods and various associated time integration methodologies for ordinary differential equations (ODEs) in time. Error in the computed solutions of initial value problems (IVPs) can be measured in the L2-norm of the solution error or Hq-norms or seminorms of errors if the theoretical solution is known. Just like boundary value problems (BVPs), the concepts of a priori and a posteriori error estimation can also be explored for IVPs also. Time accuracy or time accurate evolution implies that the computed evolution of the IVP is same as theoretical solution in as many aspects as possible. This applies to the solution as well as its spatial and time derivatives up to certain orders. In space-time coupled methods, space-time integral forms are constructed either using fundamental lemma of the calculus of variations or by using space-time residual functional (STLSP). The convergence rate estimates only hold in the asymptotic range.
