GfsOutputErrorNorm

From Gerris

GfsOutputErrorNorm is used to write the norms of the difference between a scalar field and a user-defined reference field.

The norms are written using the same formatting as for GfsOutputScalarNorm.

The syntax in parameter files is:

[ GfsOutputScalar ] {
  s = [ GfsFunction ]
  unbiased = 1
  v = Error
  w = [ GfsFunction ]
  relative = 0
}

where s is the function defining the reference field and v is used to store the difference between the scalar and reference fields (v can be left undefined in which case a temporary variable is used).

If unbiased is set to one (default is zero), the bias (i.e. mean value of the difference) is substracted before computing the norms.

The norms can be weighted using the optional parameter w.

If relative is set to one, the norms are divided by the corresponding norms of the reference solution.

Examples

• Convergence of the Poisson solver

  OutputErrorNorm { start = end } {
    awk '{print LEVEL, $5, $7, $9}' >> error 
  } { v = P } {
    s = (sin (M_PI*3.*x)*sin (M_PI*3.*y))
    unbiased = 1
  }

• Convergence with a refined circle

  OutputErrorNorm { start = end } {
    awk '{print LEVEL, $5, $7, $9}' >> error 
  } { v = P } {
    s = (sin (M_PI*3.*x)*sin (M_PI*3.*y))
    unbiased = 1
  }

• Dirichlet boundary condition

    OutputErrorNorm { start = end } {
	awk '{print LEVEL, $5, $7, $9}' >> error 
    } { v = P } {
	s = {
	    double theta = atan2 (y, x);
	    double r2 = x*x + y*y;
	    return r2*r2*cos (3.*theta);
	}
    }

• Convergence of the Godunov advection scheme

  OutputErrorNorm { start = end } { awk '{ print LEVEL " " $5 " " $7 " " $9}' } { v = T } {
    s = {
      double r2 = x*x + y*y; 
      double coeff = 20. + 20000.*r2*r2*r2*r2;
      return (1. + cos(20.*x)*cos(20.*y))*exp(-coeff*r2)/2.;
    }
  }

• Time-reversed VOF advection in a shear flow

    OutputErrorNorm { start = end } norms { v = T } {
        s = Tref v = DT
    }

• Time-reversed advection of a VOF concentration

    OutputErrorNorm { start = end } dt-LEVEL { v = T } {
	s = Tref
	v = DT
    }

    OutputErrorNorm { start = end } dt1-LEVEL { v = C } {
	s = Tref*GAUSSIAN
	v = DC
    }

    OutputErrorNorm { start = end } dt2-LEVEL { v = G } {
	s = GAUSSIAN
	v = DG
    }

• Time-reversed advection with curvature-based refinement

    OutputErrorNorm { start = end } norms { v = T } {
        s = Tref v = DT
    }

• Rotation of a straight interface

    OutputErrorNorm { istep = 1 } error { v = T } { s = Tref }

• Estimation of the numerical viscosity

  OutputErrorNorm { start = end } { awk '{ print $3,$5,$7,$9 }' > errorLEVEL.dat } {
      v = Velocity
  } { 
      s = sqrt(U0*U0+V0*V0)
      v = E
      relative = 1
  }

• Estimation of the numerical viscosity with refined box

  OutputErrorNorm { start = end } { awk '{ print $3,$5,$7,$9 }' > errorLEVEL.dat } {
      v = Velocity
  } { 
      s = sqrt(U0*U0+V0*V0)
      v = E
      relative = 1
  }

• Numerical viscosity for the skew-symmetric scheme

  OutputErrorNorm { start = end } { awk '{ print $3,$5,$7,$9 }' > errorLEVEL.dat } { 
      v = Velocity 
  } { 
      s = sqrt(U0*U0+V0*V0)
      v = E 
      relative = 1 
  }  

• Numerical viscosity for the skew-symmetric scheme with refined box

  OutputErrorNorm { start = end } { 
      awk '{ print 2**(LEVEL+1)/4+2**(LEVEL-1)*3/4,$3,$5,$7,$9 }'
  } { 
      v = Velocity 
  } { 
      s = sqrt(U0*U0 + V0*V0)
      v = E
      relative = 1 
  }

• Numerical viscosity for vorticity/streamfunction formulation

  OutputErrorNorm { start = end } { awk '{ print $3,$5,$7,$9 }' > errorLEVEL.dat } {
      v = Velocity
  } { 
      s = sqrt(U0*U0+V0*V0)
      v = E
      relative = 1
  }

• Convergence for a simple periodic problem

  OutputErrorNorm { start = end } stdout { v = U } {
    s = (1. - 2.*cos (2.*M_PI*(x - t))*sin (2.*M_PI*(y - t)))
  }

• Flow created by a cylindrical volume source

    OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' > error-LEVEL } { 
	v = Velocity 
    } {
        s = sol(x,y)
        v = E
        w = (x*x + y*y < 25.*R0*R0)
        relative = 1
    }

• Potential flow around a sphere

    OutputErrorNorm { start = end } { awk '{ print LEVEL " " $7 " " $9}' } { v = U } {
 	s = (dx("Phi") + 1.)
    }

• Poiseuille flow

    OutputErrorNorm { start = end } { awk '{print LEVEL,$5,$7,$9}' } { v = U } { 
        s = 1./2.*(1./4 - y*y) 
    }

• Bagnold flow of a granular material

    OutputErrorNorm { start = end } { awk '{print LEVEL,$5,$7,$9}' } { v = U } { 
	# Bagnold's solution
        s = UB(y)
    }

• Poiseuille flow with metric

    OutputErrorNorm { start = end } { awk '{print LEVEL,$5,$7,$9}' } { v = U } { 
        s = 1./2.*(1./4 - y*y) 
    }

• Hydrostatic balance with quadratic pressure profile

    OutputErrorNorm { istep = 1 } p { v = P } {
        s = -(cy*cy/2. + 0.5*cy) 
        unbiased = 1 
    }

• Translation of an hexagon in a uniform flow

  OutputErrorNorm { istep = 1 } {
      awk '{ printf ("%e %e %e %e\n", $3, $5, $7, $9) }' > momentumerror-ORDER
  } { v = sqrt((U - 1.)*(U - 1.) + V*V) } { s = 0. }

• Geostrophic adjustment

  OutputErrorNorm { istep = 1 } { 
      awk '{print $3/86400. " " $9; fflush (stdout); }' > e
  } { v = P/G } {
    s = ETA0*exp (-(x*x + y*y)/(R0*R0))
    unbiased = 1
    v = E
  }

• Geostrophic adjustment with Saint-Venant

  OutputErrorNorm { istep = 1 } { 
      awk '{print $3/86400. " " $9; fflush (stdout); }' > e
  } { v = P } {
    s = (H0 + ETA0*exp (-(x*x + y*y)/(R0*R0)))
    v = E
  }

• Non-linear geostrophic adjustment

  OutputErrorNorm { istep = 1 } error-OMEGA { v = P } {
      s = h0(sqrt (x*x + y*y)) 
      v = E
      unbiased = 1 
      relative = 1
  }

• Oscillations in a parabolic container

    OutputErrorNorm { istep = 1 } error-LEVEL { v = P } {
	s = Pt
	v = DP
    }

• Parabolic container with embedded solid

    OutputErrorNorm { istep = 1 } error-LEVEL { v = P } {
	s = Pt
	v = DP
    }

• Transcritical flow over a bump

    OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' } { v = H } {
	s = href.cgd
	v = EH
    }

• Lake-at-rest balance in an inclined domain with cut cells

    OutputErrorNorm { start = end } ep { v = P } {
	s = MAX(0, 0.5 - x/10.)
	unbiased = 1
	relative = 1
    }

• Lake-at-rest balance in an inclined domain with bipolar metric

    OutputErrorNorm { start = end } ep { v = P } {
	s = MAX(0, 1.26 - x/10.)
	unbiased = 1
	relative = 1
    }

• Terrain reconstruction

    OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' >> error-t } { v = T } {
	s = {
	    double r = sqrt (x*x + y*y);
	    return r*r/8.+cos(M_PI*r)/2.;
	}
	w = (fabs (x) < 3.8 && fabs (y) < 3.8)
	relative = 1
	v = Et
    }

    OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' >> error-h } { v = H0 } {
	s = {
	    double r = sqrt (x*x + y*y);
	    return r*r/8.+cos(M_PI*r)/2.;
	}
	w = (fabs (x) < 3.8 && fabs (y) < 3.8)
	relative = 1
	v = Eh
    }

• Advection of a cosine bell around the sphere

  OutputErrorNorm { istep = 1 } { awk '{ print LEVEL,$3,$5,$7,$9}' > error-LEVEL-ALPHA } { v = T } {
      s = bell_moving(x*DTR,y*DTR,t)
      v = E
      relative = 1
  }

• Poisson problem with a pure spherical harmonics solution

  OutputErrorNorm { start = end } {
    awk '{print LEVEL, $5, $7, $9}' >> error 
  } { v = P } {
    s = Sol
    v = E
    unbiased = 1
  }

• Spherical harmonics with longitude-latitude coordinates

  OutputErrorNorm { start = end } {
    awk '{print LEVEL, $5, $7, $9}' >> error 
  } { v = P } {
    s = Sol
    v = E
    unbiased = 1
  }

• Creeping Couette flow between cylinders

    OutputErrorNorm { start = end } { awk '{print LEVEL,$5,$7,$9}' } { v = V } {
	s = {
	    double r = rx/4. + 0.375;
	    // the analytical solution for the tangential velocity
	    return r*((0.5/r)*(0.5/r) - 1.)/((0.5/0.25)*(0.5/0.25) - 1.);
	}
	v = E
    }

• Creeping Couette flow between eccentric cylinders

  OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' } { v = Velocity } {
      s = {
	  double p, u, v;
	  psiuv (x, y - ECC, R1, R2, ECC, 1., 0., &u, &v, &p);
	  return sqrt (u*u + v*v);
      }
      v = EU
  }

• Flow between eccentric cylinders using bipolar coordinates

  OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' } { v = Velocity } {
      s = {
	  double p, u, v;
	  // we need to rotate coordinates by 90 degrees
	  psiuv (y, x - X2, R1, R2, ECC, 1., 0., &u, &v, &p);
	  return sqrt (u*u + v*v);
      }
      v = EU
  }

• Flow between eccentric cylinders on a stretched grid

  OutputErrorNorm { start = end } { awk '{ print LEVEL,$5,$7,$9 }' } { v = Velocity } {
      s = {
	  double p, u, v;
	  psiuv (x, y - ECC, R1, R2, ECC, 1., 0., &u, &v, &p);
	  return sqrt (u*u + v*v);
      }
      v = EU
  }

• Rossby--Haurwitz wave

    OutputErrorNorm { istart = 1 istep = 10 } eh-LEVEL { v = P/G } {
	s = p0(x*DTR,y*DTR,t)/G 
	v = EH unbiased = 1 relative = 1
    }

• Rossby--Haurwitz wave with a free surface

    OutputErrorNorm { istart = 1 istep = 10 } eh-LEVEL { v = P/G } {
	s = p0(x*DTR,y*DTR,t)/G
	v = EH unbiased = 1 relative = 1
    }

• Rossby--Haurwitz wave with Saint-Venant

    OutputErrorNorm { istep = 10 } eh-LEVEL { v = P } {
	s = p0(x*DTR,y*DTR,t)/G
	v = EH unbiased = 1 relative = 1
    }

• Relaxation of a charge bump

    OutputErrorNorm { istep = 10 } norms-SCHEME { v = Rhoe } { s = gaussbell(x,y,t) }

• Charge relaxation in an axisymmetric insulated conducting column

    OutputErrorNorm { start = end } norms-LEVEL { v = Ey } {
	s = (y < R0 ? 0 : 0.5*R0*R0*rhoinic/y)
    }

• Charge relaxation in a planar cross-section

    OutputErrorNorm { start = end } norms-LEVEL { v = sqrt(Ex*Ex + Ey*Ey) } {
	s = {
	  double r = sqrt (x*x + y*y);
          return (r < R0 ? 0 : 0.5*R0*R0*rhoinic/r);
	}
    }

• Simple example of groundwater flow following Darcy's law

	OutputErrorNorm {} { awk '{print LEVEL, $5, $7, $9}' >> p } { v = P } {
	    s = atan2 (y, x)
	}

	OutputErrorNorm {} { awk '{print LEVEL, $5, $7, $9}' >> U } { v = U } {
	    s = y / (x*x + y*y)
	}

	OutputErrorNorm {} { awk '{print LEVEL, $5, $7, $9}' >> V } { v = V } {
	    s = -x / (x*x + y*y)
	}

• Groundwater flow with piecewise constant permeability

	OutputErrorNorm {} { awk '{print LEVEL, $5, $7, $9}' >> p } { v = P } {
	    s = THETA>-M_PI/3 ? THETA : 2*THETA+M_PI/3
	}

	OutputErrorNorm {} { awk '{print LEVEL, $5, $7, $9}' >> U } { v = U } {
	    s = y / (x*x + y*y)
	}

	OutputErrorNorm {} { awk '{print LEVEL, $5, $7, $9}' >> V } { v = V } {
	    s = -x / (x*x + y*y)
	}