jwpeterson
commented
6 years ago Just wanted to note that a modified version of @YaqiWang's input file that solves the same problem without involving the "stateful" AuxKernel VariableTimeIntegrationAux behaves the same way for both Backward Euler and Crank-Nicolson, and similarly to Crank-Nicolson, LStableDirk2 is also exact for this version of the problem.
So I think the issue is definitely traceable to AuxVariables (and possibly MaterialProperties) not being updated correctly for the "multi-stage" methods (ExplicitMidpoint, LStableDirk2, explicit-tvd-rk-2, etc).
# This is a modification of Yaqi's original problem that simply solves du/dt = t
# without involving stateful AuxVariables/Postprocessors.
[Mesh]
type = GeneratedMesh
dim = 2
xmin = 0
xmax = 1
ymin = 0
ymax = 1
nx = 2
ny = 2
[]
[Variables]
[./u]
[../]
[]
[Functions]
[./forcing_fn]
type = ParsedFunction
value = 't'
[../]
[./exact_fn]
type = ParsedFunction
value = '0.5*t*t'
[../]
[]
[Kernels]
[./ie]
type = TimeDerivative
variable = u
[../]
[./cv]
type = BodyForce
variable = u
function = forcing_fn
[../]
[]
[Postprocessors]
# Estimate spatial norm of error at fixed time, ||e||_{L2}
[./l2_err]
type = ElementL2Error
variable = u
function = exact_fn
[../]
# Estimate spacetime norm ||e||_{L2, \infty}
[./max_l2_err]
type = TimeExtremeValue
value_type = max
postprocessor = l2_err
[../]
# Estimate spacetime norm ||e||_{L2, L1}
[./cumulative_l2_err]
type = CumulativeValuePostprocessor
postprocessor = l2_err
[../]
[]
[Executioner]
type = Transient
start_time = 0.0
end_time = 1.0
# dt = 1.0
# dt = 0.5
# dt = 0.25
# dt = 0.125
# dt = 0.0625
dt = 0.03125
nl_abs_tol = 1e-12
nl_rel_tol = 1e-12
# Use to set different TimeIntegrators
[./TimeIntegrator]
# type = ImplicitEuler
# type = CrankNicolson
type = LStableDirk2
[../]
[]
[Outputs]
csv = true
[]
YaqiWang
Rationale
I am able to create a simple test to show the unexpected behavior of time integrators. The test is small so I will just copy it here:
Solution of
uist*t/2. No spatial discretization error should be seen. Thus the exactintuis 0.5.Description
Backward Euler: error is
dt/2, first order; Forward Euler: error isdt/2, first order; BDF2: error is about3/4*dt^2, second order; Crank-Nicolson: error is zero; Explicit-midpoint: converges to wrong value 1; DIRK: error isdt/2, first order; Explicit-tvd-rk-2: converges to wrong value 1.Looks like BE, FE, BDF2, CN work as expected, but not the rest. I think these integrators cannot handle stateful aux variables properly. I did not try stateful material properties, but guess we will see similar issue.
Impact
Fix buggy time integrators.