Least-squares finite element processes in h, p, k mathematical and computational framework for a non-linear conservation law

This paper considers numerical simulation of time-dependent non-linear partial differential equation resulting from a single non-linear conservation law in h, p, k mathematical and computational framework in which k = (k₁, k₂) are the orders of the approximation spaces in space and time yielding global differentiability of orders (k₁ - 1) and (k₂ - 1) in space and time (hence k-version of finite element method) using space-time marching process. Time-dependent viscous Burgers equation is used as a specific model problem that has physical mechanism for viscous dissipation and its theoretical solutions are analytic. The inviscid form, on the other hand, assumes zero viscosity and as a consequence its solutions are non-analytic as well as non-unique (Russ. Math. Surv. 1962; 17(3):145-146; Russ. Math. Surv. 1960; 15(6):53-111). In references (Russ. Math. Surv. 1962; 17(3):145-146; Russ. Math. Surv. 1960; 15(6):53-111) authors demonstrated that the solutions of inviscid Burgers equations can only be approached within a limiting process in which viscosity approaches zero. Many approaches based on artificial viscosity have been published to accomplish this including more recent work on H(Div) least-squares approach (Commun. Pure Appl. Math. 1965; 18:697-715) in which artificial viscosity is a function of spatial discretization, which diminishes with progressively refined discretizations. The thrust of the present work is to point out that: (1) viscous form of the Burgers equation already has the essential mechanism of viscosity (which is physical), (2) with progressively increasing Reynolds (Re) number (thereby progressively reduced viscosity) the solutions approach that of the inviscid form, (3) it is possible to compute numerical solutions for any Re number (finite) within hpk framework and space-time least-squares processes, (4) the space-time residual functional converges monotonically and that it is possible to achieve the desired accuracy, (5) space-time, time marching processes utilizing a single space-time strip are computationally efficient. It is shown that viscous form of the Burgers equation without linearizing provides a physical and viable mechanism for approaching the solutions of inviscid form with progressively increasing Re. Numerical studies are presented and the computed solutions are compared with published work.