-- LHDI problem. The setup is described in:
--
-- Yoon, P. H., Lui, A. T. Y., & Sitnov, M. I. (2002). Generalized
-- lower-hybrid drift instabilities in current-sheet
-- equilibrium. Physics of Plasmas, 9(5), 1526. doi:10.1063/1.1466822
--
-- Yoon uses isothermal EOS, while this is the full 5-moment model.
Pi = Lucee.Pi
log = Lucee.logInfo
-- physical parameters
gasGamma = 5./3.
elcCharge = -1.0
ionCharge = 1.0
ionMass = 1.0
lightSpeed = 1.0
epsilon0 = 1.0
mu0 = 1.0
mgnErrorSpeedFactor = 1.0
-- peak number density
n0 = 1.0
plasmaBeta = 1.0 -- for Harris sheet
-- parameters (as Yoon specifies them)
M = 25.0 -- ionMass/elcMass
tau = 0.1 -- Te/Ti
R = 100.0 -- \omega_{pe}^2 / \omega_{ce}^2
vIon_vAlf = 1.0 -- ion fluid speed in terms of Alfven velocity
-- computed parameters
elcMass = ionMass/M
B0 = math.sqrt(n0*elcMass/(epsilon0*R))
vAlf = B0/math.sqrt(mu0*n0*ionMass)
omegaLH = B0*math.sqrt(ionCharge*math.abs(elcCharge)/(ionMass*elcMass))
Ti = B0^2/(2*mu0*n0*(1+tau))
Te = tau*Ti
vTe = math.sqrt(Te/elcMass)
vTi = math.sqrt(Ti/ionMass)
vIon = vIon_vAlf*vAlf
vElc = -vIon*tau
L = B0/(mu0*ionCharge*n0*(vIon-vElc))
-- cross check parameters
betaCheck = n0*(Te+Ti)/(B0^2/(2*mu0))
LCheck = 2*Ti*math.sqrt(n0*ionMass)/(ionCharge*B0*B0*vIon_vAlf)
-- domain size is based on current sheet thickness
Lx = 20*L
Ly = 20*L
-- resolution and time-stepping
NX = 500
NY = 500
cfl = 0.9
tStart = 0.0
tEnd = 100/omegaLH
nFrames = 50
log(string.format("M = %g", M))
log(string.format("B0 = %g", B0))
log(string.format("vAlf/c = %g", vAlf))
log(string.format("omegaLH = %g", omegaLH))
log(string.format("Te = %g, Ti=%g", Te, Ti))
log(string.format("vTe = %g, vTi=%g", vTe, vTi))
log(string.format("L = %g", L))
log(string.format("Lx = %g, Ly = %g", Lx, Ly))
log(string.format("L/dx = %g", L/(Lx/NX)))
log(string.format("LCheck = %g", LCheck))
log(string.format("plasmaBeta = %g", betaCheck))
log(string.format("tEnd = %g", tEnd))
------------------------------------------------
-- COMPUTATIONAL DOMAIN, DATA STRUCTURE, ETC. --
------------------------------------------------
-- decomposition object
decomp = DecompRegionCalc2D.CartGeneral {}
-- computational domain
grid = Grid.RectCart2D {
lower = {-Lx/2, -Ly/2},
upper = {Lx/2, Ly/2},
cells = {NX, NY},
decomposition = decomp,
periodicDirs = {0},
}
-- solution
q = DataStruct.Field2D {
onGrid = grid,
numComponents = 18,
ghost = {2, 2},
}
-- solution after update along X (ds algorithm)
qX = DataStruct.Field2D {
onGrid = grid,
numComponents = 18,
ghost = {2, 2},
}
-- final updated solution
qNew = DataStruct.Field2D {
onGrid = grid,
numComponents = 18,
ghost = {2, 2},
}
-- duplicate copy in case we need to take the step again
qDup = DataStruct.Field2D {
onGrid = grid,
numComponents = 18,
ghost = {2, 2},
}
qNewDup = DataStruct.Field2D {
onGrid = grid,
numComponents = 18,
ghost = {2, 2},
}
-- aliases to various sub-systems
elcFluid = q:alias(0, 5)
ionFluid = q:alias(5, 10)
emField = q:alias(10, 18)
elcFluidX = qX:alias(0, 5)
ionFluidX = qX:alias(5, 10)
emFieldX = qX:alias(10, 18)
elcFluidNew = qNew:alias(0, 5)
ionFluidNew = qNew:alias(5, 10)
emFieldNew = qNew:alias(10, 18)
-----------------------
-- INITIAL CONDITION --
-----------------------
-- initial conditions
function init(x,y,z)
-- The setup is same as in Yoon's paper (see reference on top of
-- script). The current sheet thickness can not be specified
-- explicity, but is computed from equilibrium.
local kx = 2*Pi/Lx
local ky = 2*Pi/Ly
local ypert = 0.0 --0.01*Ly*math.sin(kx*x)
--local Bz = B0*math.tanh((y+ypert)/L)
local Bz = B0*math.tanh((y+ypert)/L)*(1+ 0.05*math.random())
local sechy = 1/math.cosh((y+ypert)/L)
local nb = 0.02*n0 --1.e-3*n0
local n = n0*sechy^2 + nb
local rhoe = n*elcMass
local xmome = rhoe*vElc
local ere = n*Te/(gasGamma-1) + 0.5*rhoe*vElc^2
local rhoi = n*ionMass
local xmomi = rhoi*vIon
local eri = n*Ti/(gasGamma-1) + 0.5*rhoi*vIon^2
return rhoe, xmome, 0.0, 0.0, ere, rhoi, xmomi, 0.0, 0.0, eri, 0.0, 0.0, 0.0, 0.0, 0.0, Bz, 0.0, 0.0
end
------------------------
-- Boundary Condition --
------------------------
-- boundary applicator objects for fluids and fields
bcElcCopy = BoundaryCondition.Copy { components = {0, 4} }
bcElcWall = BoundaryCondition.ZeroNormal { components = {1, 2, 3} }
bcIonCopy = BoundaryCondition.Copy { components = {5, 9} }
bcIonWall = BoundaryCondition.ZeroNormal { components = {6, 7, 8} }
bcElcFld = BoundaryCondition.ZeroTangent { components = {10, 11, 12} }
bcMgnFld = BoundaryCondition.ZeroNormal { components = {13, 14, 15} }
bcPot = BoundaryCondition.Copy { components = {16, 17}, fact = {-1, -1} }
--FIXME: fact in bcPot
-- create boundary condition object
function createBc(myDir, myEdge)
local bc = Updater.Bc2D {
onGrid = grid,
-- boundary conditions to apply
boundaryConditions = {
bcElcCopy, bcElcWall,
bcIonCopy, bcIonWall,
bcElcFld, bcMgnFld, bcPot,
},
-- direction to apply
dir = myDir,
-- edge to apply on
edge = myEdge,
}
return bc
end
-- create updaters to apply boundary conditions
bcBottom = createBc(1, "lower")
bcTop = createBc(1, "upper")
-- function to apply boundary conditions to specified field
function applyBc(fld, tCurr, myDt)
for i,bc in ipairs({bcBottom, bcTop}) do
bc:setOut( {fld} )
bc:advance(tCurr+myDt)
end
-- sync ghost cells
fld:sync()
end
----------------------
-- EQUATION SOLVERS --
----------------------
-- regular Euler equations
elcEulerEqn = HyperEquation.Euler {
gasGamma = gasGamma,
}
ionEulerEqn = HyperEquation.Euler {
gasGamma = gasGamma,
}
-- (Lax equations are used to fix negative pressure/density)
elcEulerLaxEqn = HyperEquation.Euler {
gasGamma = gasGamma,
numericalFlux = "lax",
}
ionEulerLaxEqn = HyperEquation.Euler {
gasGamma = gasGamma,
numericalFlux = "lax",
}
maxwellEqn = HyperEquation.PhMaxwell {
lightSpeed = lightSpeed,
elcErrorSpeedFactor = 0.0,
mgnErrorSpeedFactor = mgnErrorSpeedFactor
}
-- ds solvers for regular Euler equations along X
elcFluidSlvrDir0 = Updater.WavePropagation2D {
onGrid = grid,
equation = elcEulerEqn,
-- one of no-limiter, min-mod, superbee,
-- van-leer, monotonized-centered, beam-warming
limiter = "monotonized-centered",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {0} -- directions to update
}
ionFluidSlvrDir0 = Updater.WavePropagation2D {
onGrid = grid,
equation = ionEulerEqn,
limiter = "monotonized-centered",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {0}
}
maxSlvrDir0 = Updater.WavePropagation2D {
onGrid = grid,
equation = maxwellEqn,
limiter = "monotonized-centered",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {0}
}
-- ds solvers for regular Euler equations along Y
elcFluidSlvrDir1 = Updater.WavePropagation2D {
onGrid = grid,
equation = elcEulerEqn,
limiter = "monotonized-centered",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {1}
}
ionFluidSlvrDir1 = Updater.WavePropagation2D {
onGrid = grid,
equation = ionEulerEqn,
limiter = "monotonized-centered",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {1}
}
maxSlvrDir1 = Updater.WavePropagation2D {
onGrid = grid,
equation = maxwellEqn,
limiter = "monotonized-centered",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {1}
}
-- ds solvers for Lax Euler equations along X
elcLaxSlvrDir0 = Updater.WavePropagation2D {
onGrid = grid,
equation = elcEulerLaxEqn,
limiter = "zero",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {0}
}
ionLaxSlvrDir0 = Updater.WavePropagation2D {
onGrid = grid,
equation = ionEulerLaxEqn,
limiter = "zero",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {0}
}
maxLaxSlvrDir0 = Updater.WavePropagation2D {
onGrid = grid,
equation = maxwellEqn,
limiter = "zero",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {0}
}
-- ds solvers for Lax Euler equations along Y
elcLaxSlvrDir1 = Updater.WavePropagation2D {
onGrid = grid,
equation = elcEulerLaxEqn,
limiter = "zero",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {1}
}
ionLaxSlvrDir1 = Updater.WavePropagation2D {
onGrid = grid,
equation = ionEulerLaxEqn,
limiter = "zero",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {1}
}
maxLaxSlvrDir1 = Updater.WavePropagation2D {
onGrid = grid,
equation = maxwellEqn,
limiter = "zero",
cfl = cfl,
cflm = 1.1*cfl,
updateDirections = {1}
}
-- updater for source terms
sourceSlvr = Updater.ImplicitFiveMomentSrc2D {
onGrid = grid,
numFluids = 2,
charge = {elcCharge, ionCharge},
mass = {elcMass, ionMass},
epsilon0 = epsilon0,
-- linear solver to use: one of partialPivLu or colPivHouseholderQr
linearSolver = "partialPivLu",
hasStaticField = false,
}
elcIonMomRelax = Updater.TwoFluidMomentumRelaxation2D {
onGrid = grid,
electronIonCollisionFrequency = 0.0,
frictionFactor = 0.5
}
-- function to update source terms
function updateSource(elcIn, ionIn, emIn, tCurr, t)
sourceSlvr:setOut( {elcIn, ionIn, emIn} )
sourceSlvr:setCurrTime(tCurr)
sourceSlvr:advance(t)
end
-- function to update the fluid and field using dimensional splitting
function updateFluidsAndField(tCurr, t)
local myStatus = true
local myDtSuggested = 1e3*math.abs(t-tCurr)
local useLaxSolver = False
-- X-direction updates
for i,slvr in ipairs({elcFluidSlvrDir0, ionFluidSlvrDir0, maxSlvrDir0}) do
slvr:setCurrTime(tCurr)
local status, dtSuggested = slvr:advance(t)
myStatus = status and myStatus
myDtSuggested = math.min(myDtSuggested, dtSuggested)
end
if ((elcEulerEqn:checkInvariantDomain(elcFluidX) == false)
or (ionEulerEqn:checkInvariantDomain(ionFluidX) == false)) then
useLaxSolver = true
end
if ((myStatus == false) or (useLaxSolver == true)) then
return myStatus, myDtSuggested, useLaxSolver
end
-- apply BCs to intermediate update after X sweep
applyBc(qX, tCurr, t-tCurr)
-- Y-direction updates
for i,slvr in ipairs({elcFluidSlvrDir1, ionFluidSlvrDir1, maxSlvrDir1}) do
slvr:setCurrTime(tCurr)
local status, dtSuggested = slvr:advance(t)
myStatus = status and myStatus
myDtSuggested = math.min(myDtSuggested, dtSuggested)
end
if ((elcEulerEqn:checkInvariantDomain(elcFluidNew) == false)
or (ionEulerEqn:checkInvariantDomain(ionFluidNew) == false)) then
useLaxSolver = true
end
return myStatus, myDtSuggested, useLaxSolver
end
-- function to take one time-step with Euler solver
function solveTwoFluidSystem(tCurr, t)
local dthalf = 0.5*(t-tCurr)
-- update source terms
updateSource(elcFluid, ionFluid, emField, tCurr, tCurr+dthalf)
applyBc(q, tCurr, t-tCurr)
-- update fluids and fields
local status, dtSuggested, useLaxSolver = updateFluidsAndField(tCurr, t)
-- update source terms
updateSource(elcFluidNew, ionFluidNew, emFieldNew, tCurr, tCurr+dthalf)
applyBc(qNew, tCurr, t-tCurr)
return status, dtSuggested,useLaxSolver
end
-- function to update the fluid and field using dimensional splitting Lax scheme
function updateFluidsAndFieldLax(tCurr, t)
local myStatus = true
local myDtSuggested = 1e3*math.abs(t-tCurr)
for i,slvr in ipairs({elcLaxSlvrDir0, ionLaxSlvrDir0, maxLaxSlvrDir0}) do
slvr:setCurrTime(tCurr)
local status, dtSuggested = slvr:advance(t)
myStatus = status and myStatus
myDtSuggested = math.min(myDtSuggested, dtSuggested)
end
applyBc(qX, tCurr, t-tCurr)
-- Y-direction updates
for i,slvr in ipairs({elcLaxSlvrDir1, ionLaxSlvrDir1, maxLaxSlvrDir1}) do
slvr:setCurrTime(tCurr)
local status, dtSuggested = slvr:advance(t)
myStatus = status and myStatus
myDtSuggested = math.min(myDtSuggested, dtSuggested)
end
return myStatus, myDtSuggested
end
-- function to take one time-step with Lax Euler solver
function solveTwoFluidLaxSystem(tCurr, t)
local dthalf = 0.5*(t-tCurr)
-- update source terms
updateSource(elcFluid, ionFluid, emField, tCurr, tCurr+dthalf)
applyBc(q, tCurr, t-tCurr)
-- update fluids and fields
local status, dtSuggested = updateFluidsAndFieldLax(tCurr, t)
-- update source terms
updateSource(elcFluidNew, ionFluidNew, emFieldNew, tCurr, tCurr+dthalf)
applyBc(qNew, tCurr, t-tCurr)
return status, dtSuggested
end
----------------------------
-- DIAGNOSIS AND DATA I/O --
----------------------------
-- dynvector to store electron fluid energy
elcEnergy = DataStruct.DynVector { numComponents = 1 }
elcEnergyCalc = Updater.IntegrateField2D {
onGrid = grid,
-- index of cell to record
integrand = function (rho, rhou, rhov, rhow, er)
return er
end,
}
elcEnergyCalc:setIn( {elcFluid} )
elcEnergyCalc:setOut( {elcEnergy} )
-- dynvector to store ion fluid energy
ionEnergy = DataStruct.DynVector { numComponents = 1 }
ionEnergyCalc = Updater.IntegrateField2D {
onGrid = grid,
-- index of cell to record
integrand = function (rho, rhou, rhov, rhow, er)
return er
end,
}
ionEnergyCalc:setIn( {ionFluid} )
ionEnergyCalc:setOut( {ionEnergy} )
-- dynvector to EM energy
emEnergy = DataStruct.DynVector { numComponents = 1 }
emEnergyCalc = Updater.IntegrateField2D {
onGrid = grid,
-- index of cell to record
integrand = function (ex, ey, ez, bx, by, bz, e1, e2)
return 0.5*epsilon0*(ex^2+ey^2+ez^2) + 0.5/mu0*(bx^2+by^2+bz^2)
end,
}
emEnergyCalc:setIn( {emField} )
emEnergyCalc:setOut( {emEnergy} )
-- compute diagnostic
function calcDiagnostics(tCurr, myDt)
for i,diag in ipairs({elcEnergyCalc, ionEnergyCalc, emEnergyCalc}) do
diag:setCurrTime(tCurr)
diag:advance(tCurr+myDt)
end
end
-- write data to H5 files
function writeFields(frame, t)
qNew:write( string.format("q_%d.h5", frame), t )
elcEnergy:write( string.format("elcEnergy_%d.h5", frame) )
ionEnergy:write( string.format("ionEnergy_%d.h5", frame) )
emEnergy:write( string.format("emEnergy_%d.h5", frame) )
end
----------------------------
-- TIME-STEPPING FUNCTION --
----------------------------
function runSimulation(tStart, tEnd, nFrames, initDt)
local frame = 1
local tFrame = (tEnd-tStart)/nFrames
local nextIOt = tFrame
local step = 1
local tCurr = tStart
local myDt = initDt
local status, dtSuggested
local useLaxSolver = false
-- the grand loop
while true do
-- copy q and qNew in case we need to take this step again
qDup:copy(q)
qNewDup:copy(qNew)
-- if needed adjust dt to hit tEnd exactly
if (tCurr+myDt > tEnd) then
myDt = tEnd-tCurr
end
-- advance fluids and fields
if (useLaxSolver) then
-- call Lax solver if positivity violated
log (string.format(" Taking step %5d at time %6g with dt %g (using Lax solvers)", step, tCurr, myDt))
status, dtSuggested = solveTwoFluidLaxSystem(tCurr, tCurr+myDt)
useLaxSolver = false
else
log (string.format(" Taking step %5d at time %6g with dt %g", step, tCurr, myDt))
status, dtSuggested, useLaxSolver = solveTwoFluidSystem(tCurr, tCurr+myDt)
end
if (status == false) then
-- time-step too large
log (string.format(" ** Time step %g too large! Will retake with dt %g", myDt, dtSuggested))
myDt = dtSuggested
qNew:copy(qNewDup)
q:copy(qDup)
elseif (useLaxSolver == true) then
-- negative density/pressure occured
log (string.format(" ** Negative pressure or density at %8g! Will retake step with Lax fluxes", tCurr+myDt))
q:copy(qDup)
qNew:copy(qNewDup)
else
-- check if a nan occured
if (qNew:hasNan()) then
log (string.format(" ** NaN occured at %g! Stopping simulation", tCurr))
break
end
-- compute diagnostics
calcDiagnostics(tCurr, myDt)
-- copy updated solution back
q:copy(qNew)
-- write out data
if (tCurr+myDt > nextIOt or tCurr+myDt >= tEnd) then
log (string.format(" Writing data at time %g (frame %d) ...\n", tCurr+myDt, frame))
writeFields(frame, tCurr+myDt)
frame = frame + 1
nextIOt = nextIOt + tFrame
step = 0
end
tCurr = tCurr + myDt
myDt = dtSuggested
step = step + 1
-- check if done
if (tCurr >= tEnd) then
break
end
end
end -- end of time-step loop
return dtSuggested
end
----------------------------
-- RUNNING THE SIMULATION --
----------------------------
-- setup initial condition
q:set(init)
q:sync()
qNew:copy(q)
-- set input/output arrays for various solvers
elcFluidSlvrDir0:setIn( {elcFluid} )
elcFluidSlvrDir0:setOut( {elcFluidX} )
ionFluidSlvrDir0:setIn( {ionFluid} )
ionFluidSlvrDir0:setOut( {ionFluidX} )
maxSlvrDir0:setIn( {emField} )
maxSlvrDir0:setOut( {emFieldX} )
elcFluidSlvrDir1:setIn( {elcFluidX} )
elcFluidSlvrDir1:setOut( {elcFluidNew} )
ionFluidSlvrDir1:setIn( {ionFluidX} )
ionFluidSlvrDir1:setOut( {ionFluidNew} )
maxSlvrDir1:setIn( {emFieldX} )
maxSlvrDir1:setOut( {emFieldNew} )
elcLaxSlvrDir0:setIn( {elcFluid} )
elcLaxSlvrDir0:setOut( {elcFluidX} )
ionLaxSlvrDir0:setIn( {ionFluid} )
ionLaxSlvrDir0:setOut( {ionFluidX} )
maxLaxSlvrDir0:setIn( {emField} )
maxLaxSlvrDir0:setOut( {emFieldX} )
elcLaxSlvrDir1:setIn( {elcFluidX} )
elcLaxSlvrDir1:setOut( {elcFluidNew} )
ionLaxSlvrDir1:setIn( {ionFluidX} )
ionLaxSlvrDir1:setOut( {ionFluidNew} )
maxLaxSlvrDir1:setIn( {emFieldX} )
maxLaxSlvrDir1:setOut( {emFieldNew} )
-- apply BCs on initial conditions
applyBc(q, 0.0, 0.0)
applyBc(qNew, 0.0, 0.0)
-- write initial conditions
calcDiagnostics(0.0, 0.0)
writeFields(0, 0.0)
initDt = 1.0
runSimulation(tStart, tEnd, nFrames, initDt)