eigenv.c

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#include "pluto.h"

/* ********************************************************************* */
void MaxSignalSpeed (double **v, double *cs2, 
                     double *cmin, double *cmax, int beg, int end)
/*
 *
 *    Defines the maximum and minimum propagation speeds
 *    for the RHD equations.
 *
 *    Requires:   SOUND_SPEED2
 *
 *********************************************************************** */
{
  int      i;
  double   vx, vt2, vel2;
  double   sroot, *q;

  for (i = beg; i <= end; i++) {

    q = v[i];
    
    vx   = q[VXn];
    vt2  = EXPAND(0.0, + q[VXt]*q[VXt], + q[VXb]*q[VXb]);
    vel2 = vx*vx + vt2;

    sroot = sqrt(cs2[i]*(1.0 - vx*vx - vt2*cs2[i])*(1.0 - vel2)); 
   
    cmax[i] = (vx*(1.0 - cs2[i]) + sroot)/(1.0 - vel2*cs2[i]);
    cmin[i] = (vx*(1.0 - cs2[i]) - sroot)/(1.0 - vel2*cs2[i]);

    g_maxMach = MAX(fabs(vx)/sqrt(cs2[i]), g_maxMach);
  }
}

/* ******************************************************************** */
void PrimEigenvectors (double *q, double cs2, double h, double lambda[],
                       double **LL, double **RR)
/*
 * 
 *    Provide eigenvectors eigenvalues of the relativistic
 *    equations in primitive form.
 *
 *
 * INPUT:
 *
 *   q:    a vector of primitive quantities
 * cs2:    sound speed
 *   h:    enthalpy
 *  
 * OUTPUT:
 *
 *   lambda:   returns a vector containing the eigenvalues
 *   LL,RR :   return matrices containing the left and right 
 *             eigenvectors; the column of RR are the right 
 *             eigenvectors, while the row of LL are the left
 *             eigenvectors.
 *
 *  NOTE: only non-zero components of LL and RR are computed; 
 *        RR and LL must be initialized to zero outside.
 *
 *********************************************************************** */
{
  int    nv, kk,ii,jj, ll;
  real   cs, g, g2, g3, eta;
  real   D, g2_1d;
  real   a,b, r2, r3, r4;
  real   u, v, w, rp,rm, v2tan;
#if CHECK_EIGENVECTORS == YES  
  real Aw1[NFLX], Aw0[NFLX], AA[NFLX][NFLX];
#endif

  #if RECONSTRUCT_4VEL

  /* --------------------------------------------
      compute Lorentz factor
     -------------------------------------------- */
     
   g  = EXPAND(q[VXn]*q[VXn], + q[VXt]*q[VXt], + q[VXb]*q[VXb]);
   g  = sqrt(1.0 + g);
   EXPAND(u = q[VXn]/g; ,
          v = q[VXt]/g; ,
          w = q[VXb]/g;)
  #else
   EXPAND(u = q[VXn]; ,
          v = q[VXt]; ,
          w = q[VXb];)
  #endif

/* u,v,w  are the three dimensional velocities  */

  cs    = sqrt(cs2);
  v2tan = EXPAND(0.0, + v*v, + w*w);
  eta   = sqrt(1.0 - u*u - cs2*v2tan);
  g2_1d = 1.0/(1.0 - u*u);
  g2    = u*u + v2tan;
  a     = 1.0 - g2*cs2; 
  g2    = 1.0/(1.0 - g2);
  g     = sqrt(g2);
  D     = q[RHO]*g;                  /* Lab - Density */
  rm    = u*(1.0 - cs2) - cs*eta/g;
  rp    = u*(1.0 - cs2) + cs*eta/g;

/*   Define eigenvalues   */

  lambda[0] = rm/a;
  lambda[1] = rp/a;
  for (kk = 2; kk < NFLX; kk++) lambda[kk] = u;

/* --------------------------------------------------
    Equivalent to 
    
     rp = -cs*(g*eta*u + cs)/(g*D*(1.0 - u*u));
     rm =  cs*(g*eta*u - cs)/(g*D*(1.0 - u*u));
   -------------------------------------------------- */

  rp = cs*rp/(q[RHO]*(u*cs - g*eta));
  rm = cs*rm/(q[RHO]*(u*cs + g*eta)); 

  a = cs*eta/D;
  b = h*cs2;

  #if RECONSTRUCT_4VEL == NO

  /* ======================================================
            RIGHT EIGENVECTORS, for three-vel
     ====================================================== */

  /*       lambda = u - c       */

   RR[RHO][0]  = 1.0;
   EXPAND(RR[VXn][0] = -a;           ,
          RR[VXt][0] = v*rm;  ,
          RR[VXb][0] = w*rm;)
   RR[PRS][0] = b;

  /*       lambda = u + c        */

   RR[RHO][1] = 1.0;
   EXPAND(RR[VXn][1] = a;    ,
          RR[VXt][1] = v*rp; ,
          RR[VXb][1] = w*rp;)
   RR[PRS][1] = b;

  /*       lambda = u        */

   EXPAND(RR[RHO][2] = 1.0; ,
          RR[VXt][3] = 1.0; ,
          RR[VXb][4] = 1.0;)

  /* ===================================================
                  LEFT EIGENVECTORS
     =================================================== */

   a = 0.5/a;
   b = 0.5/b;

  /*       lambda = u - c       */

   LL[0][VXn] = -a;
   LL[0][PRS] =  b;

  /*       lambda = u + c       */

   LL[1][VXn] = a;
   LL[1][PRS] = b;

  /*       lambda = u      */

   LL[2][RHO] =  1.0;
   LL[2][PRS] = -2.0*b;

   #if COMPONENTS > 1

   /*       lambda = u       */

    LL[3][VXn] = u*v*g2_1d; 
    LL[3][VXt] = 1.0;                  
    LL[3][PRS] = v*g2_1d/(g2*q[RHO]*h);

   #endif
   #if COMPONENTS > 2

  /*       lambda = u       */

    LL[4][VXn] = u*w*g2_1d; 
    LL[4][VXb] = 1.0;
    LL[4][PRS] = w*g2_1d/(g2*q[RHO]*h);

   #endif

  #elif RECONSTRUCT_4VEL == YES

/* ======================================================
                RIGHT EIGENVECTORS for four-vel
   ====================================================== */

   g3 = g2*g;

/*       lambda = u - c       */

   EXPAND(r2 = -(g + g3*u*u)*a + g3*u*rm*v2tan;  ,
          r3 = v*(-g3*u*a + rm*(g + g3*v2tan));  ,
          r4 = w*(-g3*u*a + rm*(g + g3*v2tan)); )

   RR[RHO][0] = 1.0;
   EXPAND(RR[VXn][0] = r2;  ,
          RR[VXt][0] = r3;  ,
          RR[VXb][0] = r4;)
   RR[ENG][0] = b;

/*       lambda = u + c        */

   EXPAND(r2 =  (g + g3*u*u)*a + g3*u*rp*v2tan;  ,
          r3 = v*(g3*u*a + rp*(g + g3*v2tan));   ,
          r4 = w*(g3*u*a + rp*(g + g3*v2tan)); )

   RR[RHO][1] = 1.0;
   EXPAND(RR[VXn][1] = r2;  ,
          RR[VXt][1] = r3;  ,
          RR[VXb][1] = r4;)
   RR[ENG][1] = b;

/*       lambda = u        */

   RR[RHO][2] = 1.0;

   #if COMPONENTS > 1

/*       lambda = u        */

    EXPAND(RR[VXn][3] = g3*u*v;      ,
           RR[VXt][3] = g + g3*v*v;  ,
           RR[VXb][3] = g3*v*w;)

   #endif
   #if COMPONENTS > 2

/*       lambda = u        */

    RR[VXn][4] = g3*u*w;  
    RR[VXt][4] = g3*v*w;  
    RR[VXb][4] = g + g3*w*w;

   #endif


/* ===================================================
                  LEFT EIGENVECTORS
   =================================================== */

/*       lambda = u - c       */

   EXPAND(LL[0][VXn] =-0.5*(1.0 - u*u)/a/g;  ,
          LL[0][VXt] = 0.5*u*v/a/g;    ,
          LL[0][VXb] = 0.5*u*w/a/g;)
   LL[0][ENG] = 0.5/b;

/*       lambda = u + c       */

   EXPAND(LL[1][VXn] =  0.5*(1.0 - u*u)/a/g; ,
          LL[1][VXt] = -0.5*u*v/a/g;      ,
          LL[1][VXb] = -0.5*u*w/a/g;)
   LL[1][ENG] = 0.5/b;

/*       lambda = u      */

   LL[2][RHO] = 1.0;
   LL[2][ENG]  =-1.0/b;

   #if COMPONENTS > 1

/*       lambda = u       */

    EXPAND(LL[3][VXn] = -0.5*v*(rp - rm)/a/g/g2_1d - u*v/g;  ,
           LL[3][VXt] =  0.5*v*v*(rp - rm)*u/a/g + (1.0 - v*v)/g;   ,
           LL[3][VXb] =  0.5*v*(rp - rm)*u*w/a/g - v*w/g;)
    LL[3][ENG] = -0.5*v*(rm + rp)/b;
   #endif
   #if COMPONENTS > 2

/*       lambda = u       */

    LL[4][VXn] = -0.5*w*(rp - rm)/a/g/g2_1d - u*w/g;  
    LL[4][VXt] =  0.5*v*(rp - rm)*u*w/a/g - v*w/g;
    LL[4][VXb] =  0.5*w*w*(rp - rm)*u/a/g + (1.0 - w*w)/g;
    LL[4][ENG] = -0.5*w*(rp + rm)/b;

   #endif
  #endif

/* -----------------------------------------
         Check eigenvectors consistency  
   ----------------------------------------- */

#if CHECK_EIGENVECTORS == YES

  for (ii = 0; ii <NFLX; ii++){
  for (jj = 0; jj <NFLX; jj++){
    AA[ii][jj] = 0.0;
    for (kk = 0; kk <NFLX; kk++){
    for (ll = 0; ll <NFLX; ll++){
      AA[ii][jj] += RR[ii][kk]*(kk==ll)*lambda[kk]*LL[ll][jj];
    }}
  }}

  PRIM_RHS (q, q, cs2, h, Aw0);
  for (ii = 0; ii <NFLX; ii++){
    Aw1[ii] = 0.0;
    for (jj = 0; jj <NFLX; jj++){
      Aw1[ii] += AA[ii][jj]*q[jj];
    }
  }

  a = 0.0; 
  for (ii = 0; ii <NFLX; ii++){
    a += fabs(Aw1[ii] - Aw0[ii]);
  }

  if (a > 1.e-1) {
    print ("! Eigenvectors not consistent in EIGENV%12.6e\n",a);
    for (ii = 0; ii < NFLX; ii++){
      for (jj = 0; jj < NFLX; jj++){
        print ("%12.6e   ",AA[ii][jj]);
      }
      print("\n");
    }


    print ("Aw0:  ");
    for (ii = 0; ii < NFLX; ii++){
      print ("%12.6e ,  ",Aw0[ii]);
    }
    print ("\nAw1:  ");
    for (ii = 0; ii < NFLX; ii++){
      print ("%12.6e ,  ",Aw1[ii]);
    }
    QUIT_PLUTO(1);
  }

/*  Check eigenvectors orthonormality   */

  for (ii = 0; ii <NFLX;ii++){
    for (jj = 0; jj <NFLX;jj++){
      a = 0.0;
      for (kk = 0; kk <NFLX;kk++){
        a += LL[kk][ii]*RR[jj][kk];
      }
      if (ii==jj && fabs(a-1.0)>1.e-8) {
        print ("! Eigenvectors not orthogonal!  %d  %d  %12.6e \n",ii,jj,a);
        print ("NSweep: %d\n",g_dir);
        QUIT_PLUTO(1);
      }
      if(ii!=jj && fabs(a)>1.e-8) {
        print ("! Eigenvectors not orthogonal (2) %d  %d  %12.6e !\n",ii,jj,a);
        print ("NSweep: %d\n",g_dir);
        QUIT_PLUTO(1);
      }
    } 
  }

#endif
}


/* ***************************************************************** */
void PrimToChar (double **L, double *v, double *w)
/*
 *
 *  Compute the matrix-vector product between the
 *  the L matrix (containing primitive left eigenvectors) 
 *  and the vector v. 
 *  For efficiency purpose, multiplication is done 
 *  explicitly, so that only nonzero entries
 *  of the left primitive eigenvectors are considered.
 *  
 *
 ****************************************************************** */
{
  int nv;

  #if RECONSTRUCT_4VEL == NO

   w[0] = L[0][VXn]*v[VXn] + L[0][PRS]*v[PRS];
   w[1] = L[1][VXn]*v[VXn] + L[1][PRS]*v[PRS];
   EXPAND( w[2] = v[RHO] + L[2][PRS]*v[PRS];                    ,
           w[3] = L[3][VXn]*v[VXn] + v[VXt] + L[3][PRS]*v[PRS];   ,
           w[4] = L[4][VXn]*v[VXn] + v[VXb] + L[4][PRS]*v[PRS];)

  #elif RECONSTRUCT_4VEL == YES
 
   w[0] = EXPAND(  L[0][VXn]*v[VXn],
                 + L[0][VXt]*v[VXt],
                 + L[0][VXb]*v[VXb]) + L[0][PRS]*v[PRS];
   w[1] = EXPAND(  L[1][VXn]*v[VXn],
                 + L[1][VXt]*v[VXt],
                 + L[1][VXb]*v[VXb]) + L[1][PRS]*v[PRS];;

   w[2] = v[RHO] + L[2][PRS]*v[PRS];
   
   #if COMPONENTS > 1
    w[3] = EXPAND(  L[3][VXn]*v[VXn],
                  + L[3][VXt]*v[VXt],
                  + L[3][VXb]*v[VXb]) +  L[3][PRS]*v[PRS];

    #if COMPONENTS == 3
    w[4] = L[4][VXn]*v[VXn],
           L[4][VXt]*v[VXt],
           L[4][VXb]*v[VXb] +  L[4][PRS]*v[PRS];
    #endif
   #endif

  #endif

/* ----------------------------------------------- 
     For passive scalars, the characteristic 
     variable is equal to the primitive one, 
     since  l = r = (0,..., 1 , 0 ,....)
    ----------------------------------------------- */         

#if NSCL > 0 
  NSCL_LOOP(nv) w[nv] = v[nv];
#endif   
}