Stabilization and discontinuity-capturing parameters for space–time flow computations with finite element and isogeometric discretizations

Abstract

Stabilized methods, which have been very common in flow computations for many years, typically involve stabilization parameters, and discontinuity-capturing (DC) parameters if the method is supplemented with a DC term. Various well-performing stabilization and DC parameters have been introduced for stabilized space–time (ST) computational methods in the context of the advection–diffusion equation and the Navier–Stokes equations of incompressible and compressible flows. These parameters were all originally intended for finite element discretization but quite often used also for isogeometric discretization. The stabilization and DC parameters we present here for ST computations are in the context of the advection–diffusion equation and the Navier–Stokes equations of incompressible flows, target isogeometric discretization, and are also applicable to finite element discretization. The parameters are based on a direction-dependent element length expression. The expression is outcome of an easy to understand derivation. The key components of the derivation are mapping the direction vector from the physical ST element to the parent ST element, accounting for the discretization spacing along each of the parametric coordinates, and mapping what we have in the parent element back to the physical element. The test computations we present for pure-advection cases show that the parameters proposed result in good solution profiles.

Original form of \(\tau _{\mathrm {SUPG}}\)

The component $\tau _{\mathrm {SUGN12}}$ is given [20] as

$$\begin{aligned} \tau _{\mathrm {SUGN12}}^{-1}&= \sum _{\alpha =1}^{n_{\mathrm {ent}}} \sum _{a=1}^{n_{\mathrm {ens}}} \left| \frac{\partial N_a^\alpha }{\partial t} + \mathbf {u}\cdot \pmb {\nabla }N_a^\alpha \right| . \end{aligned}$$

(60)

Alternatively, we can split $\tau _{\mathrm {SUGN12}}$ into two [21]:

$$\begin{aligned} \tau _{\mathrm {SUGN1}}^{-1}&= \sum _{\alpha =1}^{n_{\mathrm {ent}}} \sum _{a=1}^{n_{\mathrm {ens}}} \left| \left( \mathbf {u}-\mathbf {v}\right) \cdot \pmb {\nabla }N_a^\alpha \right| \end{aligned}$$

(61)

$$\begin{aligned} \tau _{\mathrm {SUGN2}}&= \frac{\varDelta t}{2}. \end{aligned}$$

(62)

The last component is given as

$$\begin{aligned} \tau _{\mathrm {SUGN3}}^{-1}&= \frac{4\nu }{h_\mathrm {RGN}^2}. \end{aligned}$$

(63)