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solvers.py
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from numpy import *
debug = False
def HLLE(fs, qs, sl, sr, sm, phi = None):
'''
makes a proxy for a half-step flux, HLLE-like
flux of quantity q, density of quantity q, sound velocity to the left, sound velocity to the right
Sod's test passed!
'''
f1,f2,f3 = fs ; q1,q2,q3 = qs
if phi is not None:
sl *= phi ; sr *= phi
sl1 = minimum(sl, 0.) ; sr1 = maximum(sr, 0.)
ds = sr1-sl1 # see Einfeldt et al. 1991 eq. 4.4
# print(ds)
# i = input("ds")
# wreg = where((sr[1:]>=0.)&(sl[:-1]<=0.)&(ds>0.))
wreg = where(ds > 0.)
w0 = where(ds <= 0.)
# wleft=where(sr[1:]<0.) ; wright=where(sl[:-1]>0.)
fhalf1=copy(f1[1:]) ; fhalf2=copy(f2[1:]) ; fhalf3=copy(f3[1:])
if(size(wreg)>0):
fhalf1[wreg] = (((sr1*f1[:-1]-sl1*f1[1:])/ds+sl1*sr1*(q1[1:]-q1[:-1])/ds))[wreg] # classic HLLE
fhalf2[wreg] = (((sr1*f2[:-1]-sl1*f2[1:])/ds+sl1*sr1*(q2[1:]-q2[:-1])/ds))[wreg] # classic HLLE
fhalf3[wreg] = (((sr1*f3[:-1]-sl1*f3[1:])/ds+sl1*sr1*(q3[1:]-q3[:-1])/ds))[wreg] # classic HLLE
if size(w0)>0.:
wpos = where((ds <=0.) & (sm >= 0.))
wneg = where((ds <=0.) & (sm <= 0.))
if size(wpos)>0.:
fhalf1[wpos] = (f1[:-1])[wpos]
fhalf2[wpos] = (f2[:-1])[wpos]
fhalf3[wpos] = (f3[:-1])[wpos]
if size(wneg)>0.:
fhalf1[wneg] = (f1[1:])[wneg]
fhalf2[wneg] = (f2[1:])[wneg]
fhalf3[wneg] = (f3[1:])[wneg]
return fhalf1, fhalf2, fhalf3
def HLLC(fs, qs, sl, sr, sm, rho, press, phi = None):
'''
makes a proxy for a half-step flux,
following the basic framework of Toro et al. 1994
flux of quantity q, density of quantity q, sound velocity to the left, sound velocity to the right, velocity of the contact discontinuity
works for Sod
does not work well for tire
'''
f1, f2, f3 = fs ; q1, q2, q3 = qs
ds = sr - sl
v = 0. # f1/q1
nx=size(q1)
fhalf1=zeros(nx-1, dtype=double) ; fhalf2=zeros(nx-1, dtype=double) ; fhalf3=zeros(nx-1, dtype=double)
if phi is not None:
sl = sl * phi
sr = sr * phi
sm = sm
fluxleft1 = sm*(sl*q1[:-1] - f1[:-1])/(sl-sm)
fluxleft2 = (sm*(sl*q2[:-1] - f2[:-1])+sl*(press[:-1] + rho[:-1]*(sl-v[:-1])*(sm-v[:-1])))/(sl-sm)
fluxleft3 = (sm*(sl*q3[:-1] - f3[:-1])+sm*sl*(press[:-1] + rho[:-1]*(sl-v[:-1])*(sm-v[:-1])))/(sl-sm)
fluxright1 = sm*(sr*q1[1:] - f1[1:])/(sr-sm)
fluxright2 = (sm*(sr*q2[1:] - f2[1:])+sr*(press[1:] + rho[1:]*(sr-v[1:])*(sm-v[1:])))/(sr-sm)
fluxright3 = (sm*(sr*q3[1:] - f3[1:])+sm*sr*(press[1:] + rho[1:]*(sr-v[1:])*(sm-v[1:])))/(sr-sm)
wsuperleft = where(sr<=0.)
wsubleft = where((sl<0.) & (sm>0.) & (sr>0.))
wsubright = where((sl<0.) & (sm<=0.) & (sr>0.))
wsuperright = where(sl>=0.)
if(debug):
print("sr = "+str(sr.min())+" to "+str(sr.max()))
print("sl = "+str(sl.min())+" to "+str(sl.max()))
print("sm = "+str(sm.min())+" to "+str(sm.max()))
print(str(size( wsuperleft))+" + "+str(size( wsubleft))+" + "+str(size( wsubright))+
" + "+str(size( wsuperright))+" + "+str(size( where(sl>= sr)))+ " = "+str(nx-1))
j = input('j')
if(size(wsubleft)>0):
fhalf1[wsubleft] = fluxleft1[wsubleft]
fhalf2[wsubleft] = fluxleft2[wsubleft]
fhalf3[wsubleft] = fluxleft3[wsubleft]
if(size(wsubright)>0):
fhalf1[wsubright] = fluxright1[wsubright]
fhalf2[wsubright] = fluxright2[wsubright]
fhalf3[wsubright] = fluxright3[wsubright]
if(size(wsuperleft)>0): # Toro eq 29
fhalf1[wsuperleft] = (f1[1:])[wsuperleft]
fhalf2[wsuperleft] = (f2[1:])[wsuperleft]
fhalf3[wsuperleft] = (f3[1:])[wsuperleft]
if(size(wsuperright)>0): # Toro eq 30
fhalf1[wsuperright] = (f1[:-1])[wsuperright]
fhalf2[wsuperright] = (f2[:-1])[wsuperright]
fhalf3[wsuperright] = (f3[:-1])[wsuperright]
wcool = where(ds<=0.)
if(size(wcool)>0):
# print(str(size(wcool))+" cool points")
wcoolright = where((sl>0.) & (ds<=0.))
wcoolleft = where((sr<0.) & (ds<=0.))
if(size(wcoolright)>0):
fhalf1[wcoolright] = (f1[:-1])[wcoolright]
fhalf2[wcoolright] = (f2[:-1])[wcoolright]
fhalf3[wcoolright] = (f3[:-1])[wcoolright]
if(size(wcoolleft)>0):
fhalf1[wcoolleft] = (f1[1:])[wcoolleft]
fhalf2[wcoolleft] = (f2[1:])[wcoolleft]
fhalf3[wcoolleft] = (f3[1:])[wcoolleft]
return fhalf1, fhalf2, fhalf3
def HLLC1(fs, qs, sl, sr, sm, rho, press, v, phi = None):
'''
second version of HLLC, according to Fleischmann et al.(2020)
apparently, works
phi is low-Mach correction
'''
f1, f2, f3 = fs ; q1, q2, q3 = qs
# ds = sr - sl
# v = f1/q1
nx=size(q1)
# fhalf1=zeros(nx-1, dtype=double) ; fhalf2=zeros(nx-1, dtype=double) ; fhalf3=zeros(nx-1, dtype=double)
if phi is not None:
sl1 = sl * phi
sr1 = sr * phi
sm1 = sm
else:
sl1 = sl
sr1 = sr
sm1 = sm
q1star_left = q1[:-1] * (sl-v[:-1])/(sl-sm)
q2star_left = q1[:-1] * (sl-v[:-1])/(sl-sm) * sm
q3star_left = (sl-v[:-1])/(sl-sm) * (q3[:-1] + (sm-v[:-1])*(sm+(press/rho)[:-1]/(sl-v[:-1])) * q1[:-1])
q1star_right = q1[1:] * (sr-v[1:])/(sr-sm)
q2star_right = q1[1:] * (sr-v[1:])/(sr-sm) * sm
q3star_right = (sr-v[1:])/(sr-sm) * (q3[1:] + (sm-v[1:])*(sm+(press/rho)[1:]/(sr-v[1:])) * q1[1:])
# compact form from Fleischmann 2021
f1half = (1.+sign(sm1))/2. * (f1[:-1]+minimum(sl1, 0.)*(q1star_left-q1[:-1])) \
+ (1.-sign(sm1))/2. * (f1[1:]+maximum(sr1, 0.)*(q1star_right-q1[1:]))
f2half = (1.+sign(sm1))/2. * (f2[:-1]+minimum(sl1, 0.)*(q2star_left-q2[:-1])) \
+ (1.-sign(sm1))/2. * (f2[1:]+maximum(sr1, 0.)*(q2star_right-q2[1:]))
f3half = (1.+sign(sm1))/2. * (f3[:-1]+minimum(sl1, 0.)*(q3star_left-q3[:-1])) \
+ (1.-sign(sm1))/2. * (f3[1:]+maximum(sr1, 0.)*(q3star_right-q3[1:]))
if False:
gamma = 4./3.
cs = sqrt(gamma * press/rho)
rhomean = (rho[1:]+rho[:-1])/2. ; csmean = (cs[1:]+cs[:-1])/2.
f2half += (1.-phi) * rhomean * csmean / 2. * fabs(v[1:]-v[:-1])
# f1half -= phi * rhomean / 2. * fabs(v[1:]-v[:-1])
return f1half, f2half, f3half
def HLLCL(fs, qs, rho, press, v, gamma = None):
'''
Kitamura & Shima 2019
'''
f1, f2, f3 = fs ; q1, q2, q3 = qs
# ds = sr - sl
# v = f1/q1
nx=size(q1)
if gamma is None:
gamma = 4./3.
cl = sqrt(gamma * press/rho)[:-1] ; cr = sqrt(gamma * press/rho)[1:]
machl = minimum(1.0, 1.0 * sqrt((v[:-1]/cl)**2+0.01))
machr = minimum(1.0, 1.0 * sqrt((v[1:]/cr)**2+0.01))
phil = machl * (2.-machl) ; phir = machr * (2.-machr)
# cl1 = cl * phil ; cr1 = cr * phir
cl1 = cl ; cr1 = cr
sl = minimum(v[:-1]-cl1, v[1:]-cr1)
sr = maximum(v[:-1]+cl1, v[1:]+cr1)
sl1 = minimum(0., sl) ; sr1 = maximum(0., sr)
sl0 = minimum(v[:-1]-cl, v[1:]-cr) ; sr0 = maximum(v[:-1]+cl, v[1:]+cr)
al = rho[:-1] * (v[:-1]-sl) ; ar = rho[1:] * (sr-v[1:])
al0 = rho[:-1] * (v[:-1]-sl0) ; ar0 = rho[1:] * (sr0-v[1:])
sm = (al * v[:-1] + ar * v[1:]) / (al+ar) - (press[1:]-press[:-1]) / (al0+ar0)
p1 = (press[1:]+press[:-1] + al * (v[:-1]-sm) - ar * (v[1:]-sm))/2. # pstar
p2 = copy(p1) # ptilde
f1half=zeros(nx-1, dtype=double) ; f2half=zeros(nx-1, dtype=double) ; f3half=zeros(nx-1, dtype=double)
ds = sr1-sl1
wreg = where(ds > 0.)
w0 = where(ds <= 0.)
f1half[wreg] = ((sr1 * f1[:-1] - sr1 * f1[1:])/(sr1-sl1) + sr1 * sl1 * (q1[1:]-q1[:-1])/(sr1-sl1))[wreg]
f2half[wreg] = ((sr1*f2[:-1]-sl1*f2[1:])/(sr1-sl1)+sl1*sr1*(q2[1:]-q2[:-1])/(sr1-sl1))[wreg]
f3half[wreg] = ((sr1*f3[:-1]-sl1*f3[1:])/(sr1-sl1)+sl1*sr1*(q3[1:]-q3[:-1])/(sr1-sl1))[wreg]
# as in classic HLLE
if size(w0)>0.:
wpos = where((ds <=0.) & (sm >= 0.))
wneg = where((ds <=0.) & (sm <= 0.))
if size(wpos)>0.:
fhalf1[wpos] = f1[:-1]
fhalf2[wpos] = f2[:-1]
fhalf3[wpos] = f3[:-1]
if size(wneg)>0.:
fhalf1[wneg] = f1[1:]
fhalf2[wneg] = f2[1:]
fhalf3[wneg] = f3[1:]
return f1half, f2half, f3half