Higher order global differentiable evolutions of initial value problems in h, p, k mathematical and computational framework

The evolution described by many (if not most) initial value problems (IVP) exhibit higher order global differentiability features in space as well as in time. For many benchmark problems permitting theoretical solutions, the evolutions are often of class C1 in space and time. It is shown that the. use of higher order local approximations in space and time in higher order Hilbert spaces are highly meritorious: (i) in incorporating the desired physics in the computational process; (ii) in incorporating desired global differentiability features; (iii) in approaching the features-of the theoretical solution in all desired aspects in the point-wise sense upon convergence; and, (iv) in permitting the use of the strong form of the governing differential equation (GDE) in the computational processes as opposed to systems of first order differential equations obtained using auxiliary variables and auxiliary equations. The space-time coupled approach is meritorious in all aspects over space-time decoupled approaches; (i) in agreement with the true nature of the evolution; (ii) in permitting space-time integral forms, (iii) permits mathematical classification of the space-time differential operators; (iv) due to (ii), one can establish a link between the space-time integral forms and elements of Calculus,of Variations to ascertain whether the space-time integral forms from various approaches such as the Galerkin method. Galerkin method with weak form, or least-squares processes are space-time variationally consistent (STVC) or space-time variationally inconsistent (STVIC); (v) due to (iv), one can establish whether the resulting Computational processes are unconditionally non-degenerate or if non-degeneracy is only conditional; (vi) STVC eliminates all issues of stability for IVP. In many IVP the initial conditions (IC) and/or boundary conditions (BC) or the other data may contain discontinuities. In such cases it is not possible to incorporate these features in the computational process correctly when one uses C-0 local approximations in space and time or C-0 local approximations non-concurrently as in space-time, decoupled approaches. It is shown that when the space-time local approximations are in higher order scalar product spaces, the descriptions of discontinuous BC or IC (or other data) in such spaces approach the true desired behavior with progressively increasing orders of the approximation space. This feature is only possible in higher order approximation spaces and permits one to maintain the smooth nature of the solutions in computations while still approaching the desired behavior in the limiting case. Simple time-dependent Couette flow of isothermal. incompressible. Newtonian fluid is used as a model problem to demonstrate the highly meritorious features of the proposed approach.