stabilisation.rnc

nu_bar_scheme = (
   ## The stabilisation parameter nu_bar is given by:
   ##
   ##   nu_bar = nu_scale xi u h
   ##
   ## With this scheme the optimal value for xi is chosen, as given by
   ## equation 2.49 in Donea & Huerta (2003):
   ##
   ##   xi = coth(pe) - 1.0 / pe
   ##
   ## Note that is potentially expensive - for small (<~10) Peclet numbers this
   ## involves evaluating hyperbolic tangents at quadrature points.
   element nu_bar_optimal {
      comment
   }|
   ## The stabilisation parameter nu_bar is given by:
   ##
   ##   nu_bar = nu_scale xi u h
   ##
   ## With this scheme a near-optimal (but more computationally efficient)
   ## value for xi is chosen, as given by equation 2.29 in Donea & Huerta
   ## (2003):
   ##
   ##   xi = pe / 3.0 if |pe| <= 3.0
   ##          sign(pe) otherwise
   element nu_bar_doubly_asymptotic {
      comment
   }|
   ## The stabilisation parameter nu_bar is given by:
   ##
   ##   nu_bar = nu_scale xi u h
   ##
   ## With this scheme a near-optimal (but more computationally efficient)
   ## value for xi is chosen, as given by equation 3.3.2 in Brookes and
   ## Hughes, Computer Methods in Applied Mechanics and Engineering 32 (1982)
   ## 199-259.
   ##
   ##   xi = -1 - 1 / pe  if pe < -1
   ##          0 if -1 <= pe <= 1
   ##          1 + 1 / pe if pe > 1
   ##
   ## This is the approximation for xi used by legacy_continuous_galerkin.
   element nu_bar_critical_rule {
      comment
   }|
   ## The stabilisation parameter nu_bar is given by:
   ##
   ##   nu_bar = nu_scale xi u h
   ##
   ## With this scheme xi is set equal to:
   ##
   ##   xi = sign(pe)
   ##
   ## This is sub-optimal, but more computationally efficient than other
   ## approximations for the stabilisation parameter. For simulations with no
   ## diffusivity, all other approximations for xi are equivalent to this one.
   element nu_bar_unity {
      comment
   }
)

nu_scale_0.5 = (
   ## nu_bar scale factor. This controls the amount of streamline upwinding.
   ##
   ## 0.5 - factor given in Donea and Huerta
   element nu_scale {
      attribute name { "0.5" },
      element real_value {
        attribute rank { "0" },
        attribute shape { "1" },
        "0.5"
      },
      comment
   }
)
nu_scale_1_over_sqrt_15 = (
   ## nu_bar scale factor. This controls the amount of streamline upwinding.
   ##
   ## 1/sqrt(15) - factor that maximises phase accuracy
   ## in 1D transient pure advection with SUPG. See Raymond and
   ## Garder, Monthly weather review 104 (1976)
   ## 1583-1590 and Brookes and Hughes, Computer Methods
   ## in Applied Mechanics and Engineering 32 (1982)
   ## 199-259.
   element nu_scale {
      attribute name { "1_over_sqrt_15" },
      element real_value {
        attribute rank { "0" },
        attribute shape { "1" },
        "0.25819889"
      },
      comment
   }
)
nu_scale_unity = (
   ## nu_bar scale factor. This controls the amount of streamline upwinding.
   ##
   ## 1.0 - no scale factor
   element nu_scale {
      attribute name { "unity"},
      element real_value {
        attribute rank { "0" },
        attribute shape { "1" },
        "1.0"
      },
      comment
   }
)
nu_scale_custom = (
   ## nu_bar scale factor. This controls the amount of streamline upwinding.
   ##
   ## Custom scale factor.
   element nu_scale {
      attribute name { "custom" },
      real,
      comment
   }
)

nu_scale_su = (
   nu_scale_0.5|
   nu_scale_unity|
   nu_scale_custom
)

nu_scale_supg = (
   nu_scale_0.5|
   nu_scale_1_over_sqrt_15|
   nu_scale_unity|
   nu_scale_custom
)

no_stabilisation = (
   ## No stabilisation
   element no_stabilisation {
      comment
   }
)

su_stabilisation = (
   ## Add a simple streamline upwind term to the advection term (if being used).
   ##
   ## This implements equation 2.52 in Donea & Huerta (2003):   
   ##
   ##   /      nu
   ##   | ----------- (U_nl\dot grad N_j)(U_nl\dot grad N_i)
   ##   / ||U_nl||**2
   element streamline_upwind {
      nu_bar_scheme,
      nu_scale_su,
      comment
   }
)

supg_stabilisation = (
   ## SUPG stabilisation, as in equation 2.51 of Donea and
   ## Huerta (2003). This only acts to stabilise if advection
   ## is not integrated by parts.
   ##
   ## <b>Under testing.</b>
   element streamline_upwind_petrov_galerkin {
      nu_bar_scheme,
      nu_scale_supg
   }
)

advection_stabilisation_options = (
   ## Stabilisation options for the galerkin discretisation
   element stabilisation{
      (
         no_stabilisation|
         su_stabilisation|
         supg_stabilisation
      )
   }
)