From b14daebfd0e56b899ce3621a2831c855ba96bce9 Mon Sep 17 00:00:00 2001
From: Lorenzo Cipriani <lorenzo@DESKTOP-L6EDCMR>
Date: Tue, 19 Sep 2023 16:01:57 +0200
Subject: [PATCH] Fixed atan initial data

---
 src/initial_data.f90 | 55 ++++++++++++++++++++++++++++++++------------
 1 file changed, 40 insertions(+), 15 deletions(-)

diff --git a/src/initial_data.f90 b/src/initial_data.f90
index d91380f..e8f95b2 100644
--- a/src/initial_data.f90
+++ b/src/initial_data.f90
@@ -9,13 +9,12 @@ subroutine initial_data(NX, x, dx, m, r0, a0, idx, u)
     real(RK), dimension(2*NX), intent(out):: u
 
     integer :: i, j
-    real(RK) :: th, Mass, heaviside, p
+    real(RK) :: th, Mass, heaviside, p, tmp
     real(RK), dimension(NX) :: rr, Marray
 
-    real(RK), parameter :: PI=4._RK*DATAN(1._RK)
+    real(RK), parameter :: PI=4._RK*DATAN(1._RK), one = 1_RK, zero = 0_RK
 
     ! Full dynamics, starting from step function
-
     if ( idx .eq. 0 ) then
 
         ! Physical values
@@ -58,29 +57,55 @@ subroutine initial_data(NX, x, dx, m, r0, a0, idx, u)
 
     ! Full dynamics, starting from atan density profile
     else if ( idx .eq. 3 ) then
-        do i = 1, NX
-            rr(i) = 3_RK * m * (PI/2_RK - atan(x(i) - r0) )  / (8_RK * PI * r0**3)
-        end do
         
-        do i = 3, NX-1
-            p = 0_RK
-            do j = 3, i
-                p = p + (rr(j-1) * x(j-1)**2 + rr(j) * x(j)**2)
-            end do
-            Marray(i) = 4_RK * PI * p * dx * 0.5_RK
+        ! Unnormalized density function
+        do i = 1, NX
+            rr(i) = PI/2_RK - atan(x(i) - r0)
         end do
 
+        ! Unnormalized mass function
+        Marray(1) = 0
         Marray(2) = 0
+        p = 0_RK
+        do i = 3, NX-1
+            p = p + (rr(i-1)*x(i-1)**2 + rr(i)*x(i)**2)
+            Marray(i) = 4*PI * p * dx * 0.5_RK
+        end do
+        Marray(NX) = Marray(NX-1)
+        
+        ! Normalized mass function
+        do i = 1, NX
+            Marray(i) =  m * Marray(i) / Marray(NX-1)
+        end do
 
         ! Physical values
         do i = 2, NX-1
             th = heaviside(r0 - x(i))
-            ! E(x)
-            u(NX+i) = - x(i)**2 / a0**2 * th - r0**2 / a0**2 * (1_RK - th)
+
+            ! Cubic interpolation around r0 to ensure epsilon is a C1 function
+            if ((x(i).le.(r0+1_RK)).and.(x(i).ge.(r0-1_RK))) then
+                u(NX+i) = x(i)**3 - (r0 + 2_RK) * x(i)**2 - (r0**2 - 1_RK) * x(i) + r0*(r0 - 1_RK)**2
+                u(NX+i) = u(NX+i) / (4_RK*a0**2)
+            else
+                u(NX+i) =  - x(i)**2 / a0**2 * th - r0**2 / a0**2 * (1_RK - th)
+            end if
+
             ! B(x)
-            u(i) = - 0.5_RK*x(i)**2 * acos(1_RK - 4_RK * Marray(i) / x(i)**3_RK - 2_RK * u(NX+i) /  x(i)**2_RK)
+            tmp = 1_RK - 4_RK * Marray(i) / x(i)**3_RK - 2_RK * u(NX+i) /  x(i)**2
+            tmp = max(-one, min(tmp, one))
+            u(i) = - 0.5_RK*x(i)**2 * acos(tmp)
+
         end do
 
+        u(2) = 0_RK
+        u(NX+2) = 0_RK
+
+        ! Ghosts
+        u(1) = u(2)
+        u(NX) = u(NX-1)
+        u(NX+1) = u(NX+2)
+        u(2*NX) = u(2*NX-1)
+
     ! Full dynamics, flat case
     else if ( idx .eq. 4 ) then