# Advection scheme mess up with tensors

SP: This is not a bug. The default boundary conditions for scalar and tensor fields are different. They must be different because scalar, vector and tensor fields do not have the same symmetries. Setting the boundary conditions for the tensor components identically, see the code in main() below, gives the same results.

Given the tracer fields A and B related by the function B = \log A ,for example. Observe that if A obeys to the advection equation B will do same,

\displaystyle \quad \partial_t A + \mathbf{u} \cdot \nabla A = 0 \quad \Rightarrow \quad \partial_t B + \mathbf{u} \cdot \nabla B = 0

Therefore, the relationship B = \log A should hold in time.

We will check that the advection scheme do not degenerate the functional relationship as time goes by. A and/or B could be scalars and/or components of a tensor.

#define DT_MAX (0.001)
#define MU0 10. //  Viscosity
#define uwall(x,t) (8*(1+tanh(8*(t-1/2)))*sq(x)*sq(1-x))

#include "navier-stokes/centered.h"

#include "tracer.h"
scalar A[], B[];
tensor T[];
scalar * tracers = {A, T.x.x, B};

int main()
{

SP: here we set the boundary conditions for T to that of a symmetric scalar field. We must use a temporary vector v because qcc is not clever enough to recognise T.x.t.

  vector v = T.x;
v.t[top] = neumann(0);
v.t[bottom] = neumann(0);
v.t[left] = neumann(0);
v.t[right] = neumann(0);

DT = DT_MAX;
N = 64;
init_grid (N);
const face vector mus[] = {MU0,MU0};
mu = mus;
run();
}

u.t[top] = dirichlet(uwall(x,t));
u.n[top] = dirichlet(0);
u.t[bottom] = dirichlet(0);
u.n[bottom] = dirichlet(0);
u.t[left] = dirichlet(0);
u.n[left] = dirichlet(0);
u.t[right] = dirichlet(0);
u.n[right] = dirichlet(0);

event init (i = 0)
{
foreach() {
u.x[] = 0.;
A[] = (4+3*sin(2*x)*cos(2*y));
B[] = log (A[]);
T.x.x[] = log (A[]);
}
boundary ((scalar *){u, A, B, T.x.x});
}

static double energy()
{
double se = 0.;
if (u.x.face)
foreach(reduction(+:se))
se += (sq(u.x[] + u.x[1,0]) + sq(u.y[] + u.y[0,1]))/8.*sq(Delta);
else // centered
foreach(reduction(+:se))
se += (sq(u.x[]) + sq(u.y[]))/2.*sq(Delta);
return se;
}

event kinetic_energy (t += 0.01)
{
//  static FILE * fp = fopen ("kinetic", "w");
fprintf (stderr, "%g %g\n", t, energy());
//fflush(fp);
}

event profile (t = 1.8)
{

FILE * fpp = fopen("yprof", "w");
for (double y = 0; y <= 1.; y += 0.01)
fprintf (fpp, "%g %g %g %g\n", y,
log(interpolate (A, 0.5, y)),
interpolate (B, 0.5, y),
interpolate (T.x.x, 0.5, y));
fclose (fpp);

fpp = fopen("xprof", "w");
for (double x = 0; x <= 1; x += 0.01)
fprintf (fpp, "%g %g %g %g \n", x,
log(interpolate (A, x, 0.75)),
interpolate (B, x, 0.75),
interpolate (T.x.x, x, 0.75));
fclose (fpp);
}

The scalars are correctly advected. The relationship B = \log A holds at instant t=1.8 . However, if the scalar is a component of a tensor (the xx component), the relationship does not stand up, i.e \mathbf{T}_{xx} \neq \log A. It is even worst if the component were xy-component.

plot 'xprof' u 1:2 w l t 'reference scalar', 'xprof' u 1:3 t 'scalar', 'xprof' u 1:4 t 'tensor' x profile at time t=1.8 (script)