(test/tests/time_integrators/scalar/stiff.i)
# This is a linear model problem described in Frank et al, "Order
# results for implicit Runge-Kutta methods applied to stiff systems",
# SIAM J. Numerical Analysis, vol. 22, no. 3, 1985, pp. 515-534.
#
# Problems "PL" and "PNL" from page 527 of the paper:
# { dy1/dt = lambda*y1 + y2**p, y1(0) = -1/(lambda+p)
# { dy2/dt = -y2, y2(0) = 1
#
# The exact solution is:
# y1 = -exp(-p*t)/(lambda+p)
# y2 = exp(-t)
#
# According to the following paragraph from the reference above, the
# p=1 version of this problem should not exhibit order reductions
# regardless of stiffness, while the nonlinear version (p>=2) will
# exhibit order reductions down to the "stage order" of the method for
# lambda large, negative.
# Use Dollar Bracket Expressions (DBEs) to set the value of LAMBDA in
# a single place. You can also set this on the command line with
# e.g. LAMBDA=-4, but note that this does not seem to override the
# value set in the input file. This is a bit different from the way
# that command line values normally work...
# Note that LAMBDA == Y2_EXPONENT is not allowed!
# LAMBDA = -10
# Y2_EXPONENT = 2
[Mesh]
type = GeneratedMesh
dim = 2
xmin = 0
xmax = 1
ymin = 0
ymax = 1
nx = 1
ny = 1
elem_type = QUAD4
[]
[Variables]
[./y1]
family = SCALAR
order = FIRST
[../]
[./y2]
family = SCALAR
order = FIRST
[../]
[]
[ICs]
[./y1_init]
type = FunctionScalarIC
variable = y1
function = y1_exact
[../]
[./y2_init]
type = FunctionScalarIC
variable = y2
function = y2_exact
[../]
[]
[ScalarKernels]
[./y1_time]
type = ODETimeDerivative
variable = y1
[../]
[./y1_space]
type = ParsedODEKernel
variable = y1
function = '-(${LAMBDA})*y1 - y2^${Y2_EXPONENT}'
args = 'y2'
[../]
[./y2_time]
type = ODETimeDerivative
variable = y2
[../]
[./y2_space]
type = ParsedODEKernel
variable = y2
function = 'y2'
[../]
[]
[Executioner]
type = Transient
[./TimeIntegrator]
type = LStableDirk2
[../]
start_time = 0
end_time = 1
dt = 0.125
solve_type = 'PJFNK'
nl_max_its = 6
nl_abs_tol = 1.e-13
nl_rel_tol = 1.e-32 # Force nl_abs_tol to be used.
line_search = 'none'
[]
[Functions]
[./y1_exact]
type = ParsedFunction
value = '-exp(-${Y2_EXPONENT}*t)/(lambda+${Y2_EXPONENT})'
vars = 'lambda'
vals = ${LAMBDA}
[../]
[./y2_exact]
type = ParsedFunction
value = exp(-t)
[../]
[]
[Postprocessors]
[./error_y1]
type = ScalarL2Error
variable = y1
function = y1_exact
execute_on = 'initial timestep_end'
[../]
[./error_y2]
type = ScalarL2Error
variable = y2
function = y2_exact
execute_on = 'initial timestep_end'
[../]
[./max_error_y1]
# Estimate ||e_1||_{\infty}
type = TimeExtremeValue
value_type = max
postprocessor = error_y1
execute_on = 'initial timestep_end'
[../]
[./max_error_y2]
# Estimate ||e_2||_{\infty}
type = TimeExtremeValue
value_type = max
postprocessor = error_y2
execute_on = 'initial timestep_end'
[../]
[./value_y1]
type = ScalarVariable
variable = y1
execute_on = 'initial timestep_end'
[../]
[./value_y2]
type = ScalarVariable
variable = y2
execute_on = 'initial timestep_end'
[../]
[./value_y1_abs_max]
type = TimeExtremeValue
value_type = abs_max
postprocessor = value_y1
execute_on = 'initial timestep_end'
[../]
[./value_y2_abs_max]
type = TimeExtremeValue
value_type = abs_max
postprocessor = value_y2
execute_on = 'initial timestep_end'
[../]
[]
[Outputs]
csv = true
[]