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ns2de.cpp
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# include <cmath>
# include <cstdlib>
# include <ctime>
# include <fstream>
# include <iomanip>
# include <iostream>
using namespace std;
# include "ns2de.hpp"
//****************************************************************************80
void grid_2d ( int x_num, double x_lo, double x_hi, int y_num, double y_lo,
double y_hi, double x[], double y[] )
//****************************************************************************80
//
// Purpose:
//
// GRID_2D returns a regular 2D grid.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 19 January 2015
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int X_NUM, the number of X values to use.
//
// Input, double X_LO, X_HI, the range of X values.
//
// Input, int Y_NUM, the number of Y values to use.
//
// Input, double Y_LO, Y_HI, the range of Y values.
//
// Output, double X[X_NUM*Y_NUM], Y[X_NUM*Y_NUM],
// the coordinates of the grid.
//
{
int i;
int j;
double xi;
double yj;
if ( x_num == 1 )
{
for ( j = 0; j < y_num; j++ )
{
for ( i = 0; i < x_num; i++ )
{
x[i+j*x_num] = ( x_lo + x_hi ) / 2.0;
}
}
}
else
{
for ( i = 0; i < x_num; i++ )
{
xi = ( ( double ) ( x_num - i - 1 ) * x_lo
+ ( double ) ( i ) * x_hi )
/ ( double ) ( x_num - 1 );
for ( j = 0; j < y_num; j++ )
{
x[i+j*x_num] = xi;
}
}
}
if ( y_num == 1 )
{
for ( j = 0; j < y_num; j++ )
{
for ( i = 0; i < x_num; i++ )
{
y[i+j*x_num] = ( y_lo + y_hi ) / 2.0;
}
}
}
else
{
for ( j = 0; j < y_num; j++ )
{
yj = ( ( double ) ( y_num - j - 1 ) * y_lo
+ ( double ) ( j ) * y_hi )
/ ( double ) ( y_num - 1 );
for ( i = 0; i < x_num; i++ )
{
y[i+j*x_num] = yj;
}
}
}
return;
}
//****************************************************************************80
void ns2de_gnuplot ( string header, int n, double x[], double y[], double u[],
double v[], double p[], double s )
//****************************************************************************80
//
// Purpose:
//
// NS2DE_GNUPLOT writes the Navier-Stokes solution to files for GNUPLOT.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 22 July 2015
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, string HEADER, a header to be used to name the files.
//
// Input, int N, the number of evaluation points.
//
// Input, double X[N], Y[N], the coordinates of the evaluation points.
//
// Input, double U[N], V[N], P[N], the solution samples.
//
// Input, double S, a scale factor for the velocity vectors.
//
{
string command_filename;
ofstream command_unit;
string data_filename;
ofstream data_unit;
int i;
string plot_filename;
//
// Write the data file.
//
data_filename = header + "_data.txt";
data_unit.open ( data_filename.c_str ( ) );
for ( i = 0; i < n; i++ )
{
data_unit << " " << x[i]
<< " " << y[i]
<< " " << u[i]
<< " " << v[i]
<< " " << s * u[i]
<< " " << s * v[i]
<< " " << p[i] << "\n";
}
data_unit.close ( );
cout << "\n";
cout << " Data written to '" << data_filename << "'\n";
//
// Write the command file.
//
command_filename = header + "_commands.txt";
plot_filename = header + ".png";
command_unit.open ( command_filename.c_str ( ) );
command_unit << "# " << command_filename << "\n";
command_unit << "#\n";
command_unit << "set term png\n";
command_unit << "set output '" << plot_filename << "'\n";
command_unit << "#\n";
command_unit << "# Add titles and labels.\n";
command_unit << "#\n";
command_unit << "set xlabel '<--- X --->'\n";
command_unit << "set ylabel '<--- Y --->'\n";
command_unit << "set title 'Navier-Stokes velocity field'\n";
command_unit << "unset key\n";
command_unit << "#\n";
command_unit << "# Add grid lines.\n";
command_unit << "#\n";
command_unit << "set grid\n";
command_unit << "set size ratio -1\n";
command_unit << "#\n";
command_unit << "# Timestamp the plot.\n";
command_unit << "#\n";
command_unit << "set timestamp\n";
command_unit << "plot '" << data_filename
<< "' using 1:2:5:6 with vectors \\\n";
command_unit << " head filled lt 2 linecolor rgb 'blue'\n";
command_unit << "quit\n";
command_unit.close ( );
cout << " Commands written to '" << command_filename << "'\n";
return;
}
//****************************************************************************80
double r8vec_amax ( int n, double a[] )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_AMAX returns the maximum absolute value in an R8VEC.
//
// Discussion:
//
// An R8VEC is a vector of R8's.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 18 September 2005
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of entries in the array.
//
// Input, double A[N], the array.
//
// Output, double AMAX, the value of the entry
// of largest magnitude.
//
{
double amax;
int i;
amax = 0.0;
for ( i = 0; i < n; i++ )
{
if ( amax < fabs ( a[i] ) )
{
amax = fabs ( a[i] );
}
}
return amax;
}
//****************************************************************************80
double r8vec_amin ( int n, double a[] )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_AMIN returns the minimum absolute value in an R8VEC.
//
// Discussion:
//
// An R8VEC is a vector of R8's.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 18 September 2005
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of entries in the array.
//
// Input, double A[N], the array.
//
// Output, double R8VEC_AMIN, the value of the entry
// of smallest magnitude.
//
{
int i;
const double r8_huge = 1.79769313486231571E+308;
double value;
value = r8_huge;
for ( i = 0; i < n; i++ )
{
if ( fabs ( a[i] ) < value )
{
value = fabs ( a[i] );
}
}
return value;
}
//****************************************************************************80
void r8vec_linspace ( int n, double a_first, double a_last, double a[] )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_LINSPACE creates a vector of linearly spaced values.
//
// Discussion:
//
// An R8VEC is a vector of R8's.
//
// 4 points evenly spaced between 0 and 12 will yield 0, 4, 8, 12.
//
// In other words, the interval is divided into N-1 even subintervals,
// and the endpoints of intervals are used as the points.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 10 April 2014
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of entries in the vector.
//
// Input, double A_FIRST, A_LAST, the first and last entries.
//
// Output, double A[N], a vector of linearly spaced data.
//
{
int i;
if ( n == 1 )
{
a[0] = ( a_first + a_last ) / 2.0;
}
else
{
for ( i = 0; i < n; i++ )
{
a[i] = ( ( double ) ( n - 1 - i ) * a_first
+ ( double ) ( i ) * a_last )
/ ( double ) ( n - 1 );
}
}
return;
}
//****************************************************************************80
double r8vec_max ( int n, double r8vec[] )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_MAX returns the value of the maximum element in an R8VEC.
//
// Discussion:
//
// An R8VEC is a vector of R8's.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 22 August 2010
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of entries in the array.
//
// Input, double R8VEC[N], a pointer to the first entry of the array.
//
// Output, double R8VEC_MAX, the value of the maximum element. This
// is set to 0.0 if N <= 0.
//
{
int i;
double value;
value = r8vec[0];
for ( i = 1; i < n; i++ )
{
if ( value < r8vec[i] )
{
value = r8vec[i];
}
}
return value;
}
//****************************************************************************80
double r8vec_min ( int n, double r8vec[] )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_MIN returns the value of the minimum element in an R8VEC.
//
// Discussion:
//
// An R8VEC is a vector of R8's.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 02 July 2005
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of entries in the array.
//
// Input, double R8VEC[N], the array to be checked.
//
// Output, double R8VEC_MIN, the value of the minimum element.
//
{
int i;
double value;
value = r8vec[0];
for ( i = 1; i < n; i++ )
{
if ( r8vec[i] < value )
{
value = r8vec[i];
}
}
return value;
}
//****************************************************************************80
double r8vec_norm_l2 ( int n, double a[] )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
//
// Discussion:
//
// An R8VEC is a vector of R8's.
//
// The vector L2 norm is defined as:
//
// R8VEC_NORM_L2 = sqrt ( sum ( 1 <= I <= N ) A(I)^2 ).
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 01 March 2003
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int N, the number of entries in A.
//
// Input, double A[N], the vector whose L2 norm is desired.
//
// Output, double R8VEC_NORM_L2, the L2 norm of A.
//
{
int i;
double v;
v = 0.0;
for ( i = 0; i < n; i++ )
{
v = v + a[i] * a[i];
}
v = sqrt ( v );
return v;
}
//****************************************************************************80
double *r8vec_uniform_ab_new ( int n, double a, double b, int &seed )
//****************************************************************************80
//
// Purpose:
//
// R8VEC_UNIFORM_AB_NEW returns a scaled pseudorandom R8VEC.
//
// Discussion:
//
// Each dimension ranges from A to B.
//
// This routine implements the recursion
//
// seed = ( 16807 * seed ) mod ( 2^31 - 1 )
// u = seed / ( 2^31 - 1 )
//
// The integer arithmetic never requires more than 32 bits,
// including a sign bit.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 09 April 2012
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Paul Bratley, Bennett Fox, Linus Schrage,
// A Guide to Simulation,
// Second Edition,
// Springer, 1987,
// ISBN: 0387964673,
// LC: QA76.9.C65.B73.
//
// Bennett Fox,
// Algorithm 647:
// Implementation and Relative Efficiency of Quasirandom
// Sequence Generators,
// ACM Transactions on Mathematical Software,
// Volume 12, Number 4, December 1986, pages 362-376.
//
// Pierre L'Ecuyer,
// Random Number Generation,
// in Handbook of Simulation,
// edited by Jerry Banks,
// Wiley, 1998,
// ISBN: 0471134031,
// LC: T57.62.H37.
//
// Peter Lewis, Allen Goodman, James Miller,
// A Pseudo-Random Number Generator for the System/360,
// IBM Systems Journal,
// Volume 8, Number 2, 1969, pages 136-143.
//
// Parameters:
//
// Input, int N, the number of entries in the vector.
//
// Input, double A, B, the lower and upper limits of the pseudorandom values.
//
// Input/output, int &SEED, a seed for the random number generator.
//
// Output, double R8VEC_UNIFORM_AB_NEW[N], the vector of pseudorandom values.
//
{
int i;
const int i4_huge = 2147483647;
int k;
double *r;
if ( seed == 0 )
{
cerr << "\n";
cerr << "R8VEC_UNIFORM_AB_NEW - Fatal error!\n";
cerr << " Input value of SEED = 0.\n";
exit ( 1 );
}
r = new double[n];
for ( i = 0; i < n; i++ )
{
k = seed / 127773;
seed = 16807 * ( seed - k * 127773 ) - k * 2836;
if ( seed < 0 )
{
seed = seed + i4_huge;
}
r[i] = a + ( b - a ) * ( double ) ( seed ) * 4.656612875E-10;
}
return r;
}
//****************************************************************************80
void resid_lucas ( double nu, double rho, int n, double x[], double y[],
double t, double ur[], double vr[], double pr[] )
//****************************************************************************80
//
// Purpose:
//
// RESID_LUCAS returns Lucas Bystricky residuals.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 06 March 2015
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, double NU, the kinematic viscosity.
//
// Input, double RHO, the density.
//
// Input, int N, the number of evaluation points.
//
// Input, double X[N], Y[N], the coordinates of the points.
//
// Input, double T, the time coordinate or coordinates.
//
// Output, double UR[N], VR[N], PR[N], the residuals in the U,
// V and P equations.
//
{
double dpdx;
double dpdy;
double dudt;
double dudx;
double dudxx;
double dudy;
double dudyy;
double dvdt;
double dvdx;
double dvdxx;
double dvdy;
double dvdyy;
double *f;
double *g;
double *h;
int i;
double p;
const double r8_pi = 3.141592653589793;
double u;
double v;
//
// Get the right hand sides.
//
f = new double[n];
g = new double[n];
h = new double[n];
rhs_lucas ( nu, rho, n, x, y, t, f, g, h );
//
// Form the functions and derivatives of the left hand side.
//
for ( i = 0; i < n; i++ )
{
u = - cos ( r8_pi * x[i] ) / r8_pi;
dudt = 0.0;
dudx = sin ( r8_pi * x[i] );
dudxx = r8_pi * cos ( r8_pi * x[i] );
dudy = 0.0;
dudyy = 0.0;
v = - y[i] * sin ( r8_pi * x[i] );
dvdt = 0.0;
dvdx = - r8_pi * y[i] * cos ( r8_pi * x[i] );
dvdxx = + r8_pi * r8_pi * y[i] * sin ( r8_pi * x[i] );
dvdy = - sin ( r8_pi * x[i] );
dvdyy = 0.0;
p = 0.0;
dpdx = 0.0;
dpdy = 0.0;
//
// Evaluate the residuals.
//
ur[i] = dudt + u * dudx + v * dudy
+ ( 1.0 / rho ) * dpdx - nu * ( dudxx + dudyy ) - f[i];
vr[i] = dvdt + u * dvdx + v * dvdy
+ ( 1.0 / rho ) * dpdy - nu * ( dvdxx + dvdyy ) - g[i];
pr[i] = dudx + dvdy - h[i];
}
//
// Free memory.
//
delete [] f;
delete [] g;
delete [] h;
return;
}
//****************************************************************************80
void resid_poiseuille ( double nu, double rho, int n, double x[], double y[],
double t, double ur[], double vr[], double pr[] )
//****************************************************************************80
//
// Purpose:
//
// RESID_POISEUILLE returns Poiseuille residuals.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 22 July 2015
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, double NU, the kinematic viscosity.
//
// Input, double RHO, the density.
//
// Input, int N, the number of evaluation points.
//
// Input, double X[N], Y[N], the coordinates of the points.
//
// Input, double T, the time coordinate or coordinates.
//
// Output, double UR[N], VR[N], PR[N], the residuals in the U,
// V and P equations.
//
{
double dpdx;
double dpdy;
double dudt;
double dudx;
double dudxx;
double dudy;
double dudyy;
double dvdt;
double dvdx;
double dvdxx;
double dvdy;
double dvdyy;
double *f;
double *g;
double *h;
int i;
double p;
double u;
double v;
//
// Get the right hand sides.
//
f = new double[n];
g = new double[n];
h = new double[n];
rhs_poiseuille ( nu, rho, n, x, y, t, f, g, h );
//
// Form the functions and derivatives of the left hand side.
//
for ( i = 0; i < n; i++ )
{
u = 1.0 - y[i] * y[i];
dudt = 0.0;
dudx = 0.0;
dudxx = 0.0;
dudy = - 2.0 * y[i];
dudyy = - 2.0;
v = 0.0;
dvdt = 0.0;
dvdx = 0.0;
dvdxx = 0.0;
dvdy = 0.0;
dvdyy = 0.0;
p = - 2.0 * rho * nu * x[i];
dpdx = - 2.0 * rho * nu;
dpdy = 0.0;
//
// Evaluate the residuals.
//
ur[i] = dudt + u * dudx + v * dudy
+ ( 1.0 / rho ) * dpdx - nu * ( dudxx + dudyy ) - f[i];
vr[i] = dvdt + u * dvdx + v * dvdy
+ ( 1.0 / rho ) * dpdy - nu * ( dvdxx + dvdyy ) - g[i];
pr[i] = dudx + dvdy - h[i];
}
//
// Free memory.
//
delete [] f;
delete [] g;
delete [] h;
return;
}
//****************************************************************************//
void resid_spiral ( double nu, double rho, int n, double x[], double y[],
double t, double ur[], double vr[], double pr[] )
//****************************************************************************//
//
// Purpose:
//
// RESID_SPIRAL evaluates Spiral residuals.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 30 January 2015
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Maxim Olshanskii, Leo Rebholz,
// Application of barycenter refined meshes in linear elasticity
// and incompressible fluid dynamics,
// ETNA: Electronic Transactions in Numerical Analysis,
// Volume 38, pages 258-274, 2011.
//
// Parameters:
//
// Input, double NU, the kinematic viscosity.
//
// Input, double RHO, the fluid density.
//
// Input, int N, the number of nodes.
//
// Input, double X[N], Y[N], the coordinates of nodes.
//
// Input, double T, the current time.
//
// Output, double UR[N], VR[N], PR[N], the right hand sides.
//
{
double *dpdx;
double *dpdy;
double *dudt;
double *dudx;
double *dudxx;
double *dudy;
double *dudyy;
double *dvdt;
double *dvdx;
double *dvdxx;
double *dvdy;
double *dvdyy;
double *f;
double *g;
double *h;
int i;
double *p;
double *u;
double *v;
dpdx = new double[n];
dpdy = new double[n];
dudt = new double[n];
dudx = new double[n];
dudxx = new double[n];
dudy = new double[n];
dudyy = new double[n];
dvdt = new double[n];
dvdx = new double[n];
dvdxx = new double[n];
dvdy = new double[n];
dvdyy = new double[n];
f = new double[n];
g = new double[n];
h = new double[n];
p = new double[n];
u = new double[n];
v = new double[n];
//
// Get the right hand side functions.
//
rhs_spiral ( nu, rho, n, x, y, t, f, g, h );
//
// Form the functions and derivatives for the left hand side.
//
for ( i = 0; i < n; i++ )
{
u[i] = ( 1.0 + nu * t ) * 2.0
* ( pow ( x[i], 4 ) - 2.0 * pow ( x[i], 3 ) + pow ( x[i], 2 ) )
* ( 2.0 * pow ( y[i], 3 ) - 3.0 * pow ( y[i], 2 ) + y[i] );
dudt[i] = nu * 2.0
* ( pow ( x[i], 4 ) - 2.0 * pow ( x[i], 3 ) + pow ( x[i], 2 ) )
* ( 2.0 * pow ( y[i], 3 ) - 3.0 * pow ( y[i], 2 ) + y[i] );
dudx[i] = ( 1.0 + nu * t ) * 2.0
* ( 4.0 * pow ( x[i], 3 ) - 6.0 * pow ( x[i], 2 ) + 2.0 * x[i] )
* ( 2.0 * pow ( y[i], 3 ) - 3.0 * pow ( y[i], 2 ) + y[i] );
dudxx[i] = ( 1.0 + nu * t ) * 2.0
* ( 12.0 * pow ( x[i], 2 ) - 12.0 * x[i] + 2.0 )
* ( 2.0 * pow ( y[i], 3 ) - 3.0 * pow ( y[i], 2 ) + y[i] );
dudy[i] = ( 1.0 + nu * t ) * 2.0
* ( pow ( x[i], 4 ) - 2.0 * pow ( x[i], 3 ) + pow ( x[i], 2 ) )
* ( 6.0 * pow ( y[i], 2 ) - 6.0 * y[i] + 1.0 );
dudyy[i] = ( 1.0 + nu * t ) * 2.0
* ( pow ( x[i], 4 ) - 2.0 * pow ( x[i], 3 ) + pow ( x[i], 2 ) )
* ( 12.0 * y[i] - 6.0 );
v[i] = - ( 1.0 + nu * t ) * 2.0
* ( 2.0 * pow ( x[i], 3 ) - 3.0 * pow ( x[i], 2 ) + x[i] )
* ( pow ( y[i], 4 ) - 2.0 * pow ( y[i], 3 ) + pow ( y[i], 2 ) );
dvdt[i] = - nu * 2.0
* ( 2.0 * pow ( x[i], 3 ) - 3.0 * pow ( x[i], 2 ) + x[i] )
* ( pow ( y[i], 4 ) - 2.0 * pow ( y[i], 3 ) + pow ( y[i], 2 ) );
dvdx[i] = - ( 1.0 + nu * t ) * 2.0
* ( 6.0 * pow ( x[i], 2 ) - 6.0 * x[i] + 1.0 )
* ( pow ( y[i], 4 ) - 2.0 * pow ( y[i], 3 ) + pow ( y[i], 2 ) );
dvdxx[i] = - ( 1.0 + nu * t ) * 2.0
* ( 12.0 * x[i] - 6.0 )
* ( pow ( y[i], 4 ) - 2.0 * pow ( y[i], 3 ) + pow ( y[i], 2 ) );
dvdy[i] = - ( 1.0 + nu * t ) * 2.0
* ( 2.0 * pow ( x[i], 3 ) - 3.0 * pow ( x[i], 2 ) + x[i] )
* ( 4.0 * pow ( y[i], 3 ) - 6.0 * pow ( y[i], 2 ) + 2.0 * y[i] );
dvdyy[i] = - ( 1.0 + nu * t ) * 2.0
* ( 2.0 * pow ( x[i], 3 ) - 3.0 * pow ( x[i], 2 ) + x[i] )
* ( 12.0 * pow ( y[i], 2 ) - 12.0 * y[i] + 2.0 );
p[i] = rho * y[i];
dpdx[i] = 0.0;
dpdy[i] = rho;
ur[i] = dudt[i] - nu * ( dudxx[i] + dudyy[i] )
+ u[i] * dudx[i] + v[i] * dudy[i] + dpdx[i] / rho - f[i];
vr[i] = dvdt[i] - nu * ( dvdxx[i] + dvdyy[i] )
+ u[i] * dvdx[i] + v[i] * dvdy[i] + dpdy[i] / rho - g[i];
pr[i] = dudx[i] + dvdy[i] - h[i];
}
delete [] dpdx;
delete [] dpdy;
delete [] dudt;
delete [] dudx;
delete [] dudxx;
delete [] dudy;
delete [] dudyy;
delete [] dvdt;
delete [] dvdx;