# This test solves a 1D transient heat equation
# The error is caclulated by comparing to the analytical solution
# The problem setup and analytical solution are taken from "Advanced Engineering
# Mathematics, 10th edition" by Erwin Kreyszig.
# http://www.amazon.com/Advanced-Engineering-Mathematics-Erwin-Kreyszig/dp/0470458364
# It is Example 1 in section 12.6 on page 561
[Mesh]
type = GeneratedMesh
dim = 2
nx = 40
ny = 40
xmax = 80
ymax = 80
[]
[Variables]
[./T]
[../]
[]
[Kernels]
[./HeatSource]
type = HeatSource
function = '100*sin(3.14159*x/80)*sin(3.14159*y/80)'
variable = T
[../]
[./HeatDiff]
type = HeatConduction
variable = T
[../]
[./HeatTdot]
type = HeatConductionTimeDerivative
variable = T
[../]
[]
[BCs]
[./left]
type = DirichletBC
variable = T
boundary = left
value = 0
[../]
[./right]
type = DirichletBC
variable = T
boundary = right
value = 0
[../]
[./top]
type = DirichletBC
variable = T
boundary = top
value = 0
[../]
[./bottom]
type = DirichletBC
variable = T
boundary = bottom
value = 0
[../]
[]
[Materials]
[./k]
type = GenericConstantMaterial
prop_names = 'thermal_conductivity'
prop_values = '0.95' #copper in cal/(cm sec C)
block = 0
[../]
[./cp]
type = GenericConstantMaterial
prop_names = 'specific_heat'
prop_values = '0.092' #copper in cal/(g C)
block = 0
[../]
[./rho]
type = GenericConstantMaterial
prop_names = 'density'
prop_values = '8.92' #copper in g/(cm^3)
block = 0
[../]
[]
[Postprocessors]
[./error]
type = NodalL2Error
function = '100*sin(3.14159*x/80)*exp(-0.95/(0.092*8.92)*3.14159^2/80^2*t)' #T(x,t) = 100sin(pix/L)exp(-rho/(cp k) pi^2/L^2 t)
variable = T
[../]
[]
[Executioner]
type = Transient
scheme = bdf2
nl_rel_tol = 1e-12
l_tol = 1e-6
dt = 2
end_time = 100
petsc_options_iname = '-pc_type -pc_hypre_type'
petsc_options_value = 'hypre boomeramg'
[]
[Outputs]
exodus = true
print_perf_log = true
[]