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-rw-r--r--m/McLachlan_BSSN.m91
1 files changed, 53 insertions, 38 deletions
diff --git a/m/McLachlan_BSSN.m b/m/McLachlan_BSSN.m
index 9647ccf..877a879 100644
--- a/m/McLachlan_BSSN.m
+++ b/m/McLachlan_BSSN.m
@@ -447,7 +447,7 @@ evolCalc =
Gt[ua,lb,lc], Xtn[ua], Rt[la,lb], Rphi[la,lb], R[la,lb],
Atm[ua,lb], Atu[ua,ub],
e4phi, em4phi, cdphi[la], cdphi2[la,lb], g[la,lb], detg,
- gu[ua,ub], G[ua,lb,lc], Ats[la,lb], trAts, eta,
+ gu[ua,ub], Ats[la,lb], trAts, eta,
rho, S[la], trS, fac1, fac2},
Equations ->
{
@@ -494,9 +494,6 @@ evolCalc =
detg -> detgExpr,
(* gu[ua,ub] -> 1/detg detgExpr MatrixInverse [g[ua,ub]], *)
gu[ua,ub] -> em4phi gtu[ua,ub],
- G[ua,lb,lc] -> Gt[ua,lb,lc]
- + 2 (KD[ua,lb] cdphi[lc] + KD[ua,lc] cdphi[lb]
- - gtu[ua,ud] gt[lb,lc] cdphi[ld]),
R[la,lb] -> Rt[la,lb] + Rphi[la,lb],
@@ -540,7 +537,9 @@ evolCalc =
+ addMatter (- 16 pi alpha gtu[ui,uj] S[lj]),
(* PRD 62, 044034 (2000), eqn. (11) *)
- dot[trK] -> - gu[ua,ub] CD[alpha,la,lb]
+ dot[trK] -> - em4phi ( gtu[ua,ub] ( PD[alpha,la,lb]
+ + 2 cdphi[la] PD[alpha,lb] )
+ - Xtn[ua] PD[alpha,la] )
+ alpha (Atm[ua,lb] Atm[ub,la] + (1/3) trK^2)
+ beta[ua] PDu[trK,la]
(* Equation (4.21) in Baumgarte & Shapiro (Phys. Rept. 376 (2003) 41-131) *)
@@ -548,7 +547,9 @@ evolCalc =
(* PRD 62, 044034 (2000), eqn. (12) *)
(* TODO: Should we use the Hamiltonian constraint to make Rij tracefree? *)
- Ats[la,lb] -> - CD[alpha,la,lb] + alpha R[la,lb],
+ Ats[la,lb] -> - CDt[alpha,la,lb] +
+ + 2 (PD[alpha,la] cdphi[lb] + PD[alpha,lb] cdphi[la] )
+ + alpha R[la,lb],
trAts -> gu[ua,ub] Ats[la,lb],
dot[At[la,lb]] -> + em4phi (+ Ats[la,lb] - (1/3) g[la,lb] trAts )
+ alpha (trK At[la,lb] - 2 At[la,lc] Atm[uc,lb])
@@ -594,10 +595,10 @@ evol1Calc =
Where -> InteriorNoSync,
Shorthands -> {dir[ua],
detgt, gtu[ua,ub],
- Gt[ua,lb,lc], Xtn[ua], Rt[la,lb], Rphi[la,lb], R[la,lb],
+ Gt[ua,lb,lc], Xtn[ua],
Atm[ua,lb], Atu[ua,ub],
- e4phi, em4phi, cdphi[la], cdphi2[la,lb], g[la,lb], detg, eta,
- rho, S[la], trS, fac1, fac2},
+ e4phi, em4phi, cdphi[la], eta,
+ rho, S[la], trS, fac1},
Equations ->
{
dir[ua] -> Sign[beta[ua]],
@@ -619,13 +620,35 @@ evol1Calc =
Atm[ua,lb] -> gtu[ua,uc] At[lc,lb],
Atu[ua,ub] -> Atm[ua,lc] gtu[ub,uc],
+ e4phi -> IfThen [conformalMethod, 1/phi^2, Exp[4 phi]],
+ em4phi -> 1 / e4phi,
+
(* Matter terms *)
+ (* rho = n^a n^b T_ab *)
+ rho -> addMatter
+ (1/alpha^2 (T00 - 2 beta[ui] T0[li] + beta[ui] beta[uj] T[li,lj])),
+
+ (* trS = gamma^ij T_ij *)
+ trS -> addMatter (em4phi gtu[ui,uj] T[li,lj]),
+
(* S_i = -p^a_i n^b T_ab, where p^a_i = delta^a_i + n^a n_i *)
S[li] -> addMatter (-1/alpha (T0[li] - beta[uj] T[li,lj])),
(* RHS terms *)
+ (* PRD 62, 044034 (2000), eqn. (11) *)
+ (* Note, Covariant derivatives with respect to the physical metric
+ has been replaced with derivatives with respect to the
+ conformal metric *)
+ dot[trK] -> - em4phi ( gtu[ua,ub] ( PD[alpha,la,lb]
+ + 2 cdphi[la] PD[alpha,lb] )
+ - Xtn[ua] PD[alpha,la] )
+ + alpha (Atm[ua,lb] Atm[ub,la] + (1/3) trK^2)
+ + beta[ua] PDu[trK,la]
+ (* Equation (4.21) in Baumgarte & Shapiro (Phys. Rept. 376 (2003) 41-131) *)
+ + addMatter (4 pi alpha (rho + trS)),
+
(* PRD 62, 044034 (2000), eqn. (10) *)
(* PRD 67 084023 (2003), eqn. (16) and (23) *)
dot[phi] -> IfThen [conformalMethod, 1/3 phi, -1/6] alpha trK
@@ -637,6 +660,7 @@ evol1Calc =
+ beta[uc] PDu[gt[la,lb],lc]
+ gt[la,lc] PD[beta[uc],lb] + gt[lb,lc] PD[beta[uc],la]
- (2/3) gt[la,lb] PD[beta[uc],lc],
+
(* PRD 62, 044034 (2000), eqn. (20) *)
(* PRD 67 084023 (2003), eqn (26) *)
dot[Xt[ui]] -> - 2 Atu[ui,uj] PD[alpha,lj]
@@ -654,6 +678,14 @@ evol1Calc =
eta -> BetaDriver
IfThen [r > SpatialBetaDriverRadius, SpatialBetaDriverRadius / r, 1],
+ (* dot[alpha] -> - harmonicF alpha^harmonicN trK, *)
+ (* dot[alpha] -> - harmonicF alpha^harmonicN A + Lie[alpha, beta], *)
+ dot[alpha] -> - harmonicF alpha^harmonicN (
+ (1 - LapseAdvectionCoeff) A + LapseAdvectionCoeff trK)
+ + LapseAdvectionCoeff beta[ua] PDu[alpha,la],
+
+ dot[A] -> (1 - LapseAdvectionCoeff) (dot[trK] - AlphaDriver A),
+
(* dot[beta[ua]] -> eta Xt[ua], *)
(* dot[beta[ua]] -> ShiftGammaCoeff alpha^ShiftAlphaPower B[ua], *)
dot[beta[ua]] -> + ShiftGammaCoeff B[ua]
@@ -682,7 +714,7 @@ evol2Calc =
Rt[la,lb], Rphi[la,lb], R[la,lb],
Atm[ua,lb],
e4phi, em4phi, cdphi[la], cdphi2[la,lb], g[la,lb], detg,
- gu[ua,ub], G[ua,lb,lc], Ats[la,lb], trAts,
+ gu[ua,ub], Ats[la,lb], trAts,
rho, S[la], trS, fac1, fac2},
Equations ->
{
@@ -729,33 +761,24 @@ evol2Calc =
g[la,lb] -> e4phi gt[la,lb],
(* gu[ua,ub] -> 1/detg detgExpr MatrixInverse [g[ua,ub]], *)
gu[ua,ub] -> em4phi gtu[ua,ub],
- G[ua,lb,lc] -> Gt[ua,lb,lc]
- + 2 (KD[ua,lb] cdphi[lc] + KD[ua,lc] cdphi[lb]
- - gtu[ua,ud] gt[lb,lc] cdphi[ld]),
R[la,lb] -> Rt[la,lb] + Rphi[la,lb],
(* Matter terms *)
- (* rho = n^a n^b T_ab *)
- rho -> addMatter
- (1/alpha^2 (T00 - 2 beta[ui] T0[li] + beta[ui] beta[uj] T[li,lj])),
-
(* trS = gamma^ij T_ij *)
trS -> addMatter (gu[ui,uj] T[li,lj]),
(* RHS terms *)
- (* PRD 62, 044034 (2000), eqn. (11) *)
- dot[trK] -> - gu[ua,ub] CD[alpha,la,lb]
- + alpha (Atm[ua,lb] Atm[ub,la] + (1/3) trK^2)
- + beta[ua] PDu[trK,la]
- (* Equation (4.21) in Baumgarte & Shapiro (Phys. Rept. 376 (2003) 41-131) *)
- + addMatter (4 pi alpha (rho + trS)),
-
(* PRD 62, 044034 (2000), eqn. (12) *)
(* TODO: Should we use the Hamiltonian constraint to make Rij tracefree? *)
- Ats[la,lb] -> - CD[alpha,la,lb] + alpha R[la,lb],
+ (* Note, Covariant derivatives with respect to the physical metric
+ has been replaced with derivatives with respect to the
+ conformal metric *)
+ Ats[la,lb] -> - CDt[alpha,la,lb] +
+ + 2 (PD[alpha,la] cdphi[lb] + PD[alpha,lb] cdphi[la] )
+ + alpha R[la,lb],
trAts -> gu[ua,ub] Ats[la,lb],
dot[At[la,lb]] -> + em4phi (+ Ats[la,lb] - (1/3) g[la,lb] trAts )
+ alpha (trK At[la,lb] - 2 At[la,lc] Atm[uc,lb])
@@ -764,15 +787,7 @@ evol2Calc =
- (2/3) At[la,lb] PD[beta[uc],lc]
(* Equation (4.23) in Baumgarte & Shapiro (Phys. Rept. 376 (2003) 41-131) *)
+ addMatter (- em4phi alpha 8 pi
- (T[la,lb] - (1/3) g[la,lb] trS)),
-
- (* dot[alpha] -> - harmonicF alpha^harmonicN trK, *)
- (* dot[alpha] -> - harmonicF alpha^harmonicN A + Lie[alpha, beta], *)
- dot[alpha] -> - harmonicF alpha^harmonicN (
- (1 - LapseAdvectionCoeff) A + LapseAdvectionCoeff trK)
- + LapseAdvectionCoeff beta[ua] PDu[alpha,la],
-
- dot[A] -> (1 - LapseAdvectionCoeff) (dot[trK] - AlphaDriver A)
+ (T[la,lb] - (1/3) g[la,lb] trS))
}
};
@@ -1226,7 +1241,7 @@ CreateKrancThornTT [groups, ".", BSSN,
(* matter: 0 or 1
(matter seems cheap; it should be always enabled) *)
-createCode[2, False, 4, 1];
-createCode[4, False, 4, 1];
-createCode[4, True, 4, 1];
-createCode[8, False, 4, 1];
+createCode[2, False, 3, 1];
+createCode[4, False, 3, 1];
+createCode[4, True, 3, 1];
+createCode[8, False, 3, 1];
generated by cgit v1.2.3 (git 2.39.1) at 2024-07-02 12:16:44 +0000