private static void Main()
//****************************************************************************80
//
// Purpose:
//
// MAIN is the main program for FD1D_HEAT_IMPLICIT.
//
// Discussion:
//
// FD1D_HEAT_IMPLICIT solves the 1D heat equation with an implicit method.
//
// This program solves
//
// dUdT - k * d2UdX2 = F(X,T)
//
// over the interval [A,B] with boundary conditions
//
// U(A,T) = UA(T),
// U(B,T) = UB(T),
//
// over the time interval [T0,T1] with initial conditions
//
// U(X,T0) = U0(X)
//
// The code uses the finite difference method to approximate the
// second derivative in space, and an implicit backward Euler approximation
// to the first derivative in time.
//
// The finite difference form can be written as
//
// U(X,T+dt) - U(X,T) ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) )
// ------------------ = F(X,T) + k * ------------------------------------
// dt dx * dx
//
// so that we have the following linear system for the values of U at time T+dt:
//
// - k * dt / dx / dx * U(X-dt,T+dt)
// + ( 1 + 2 * k * dt / dx / dx ) * U(X, T+dt)
// - k * dt / dx / dx * U(X+dt,T+dt)
// = dt * F(X, T+dt)
// + U(X, T)
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 31 May 2009
//
// Author:
//
// John Burkardt
//
{
int i;
int j;
Console.WriteLine("");
Console.WriteLine("FD1D_HEAT_IMPLICIT");
Console.WriteLine("");
Console.WriteLine(" Finite difference solution of");
Console.WriteLine(" the time dependent 1D heat equation");
Console.WriteLine("");
Console.WriteLine(" Ut - k * Uxx = F(x,t)");
Console.WriteLine("");
Console.WriteLine(" for space interval A <= X <= B with boundary conditions");
Console.WriteLine("");
Console.WriteLine(" U(A,t) = UA(t)");
Console.WriteLine(" U(B,t) = UB(t)");
Console.WriteLine("");
Console.WriteLine(" and time interval T0 <= T <= T1 with initial condition");
Console.WriteLine("");
Console.WriteLine(" U(X,T0) = U0(X).");
Console.WriteLine("");
Console.WriteLine(" A second order difference approximation is used for Uxx.");
Console.WriteLine("");
Console.WriteLine(" A first order backward Euler difference approximation");
Console.WriteLine(" is used for Ut.");
const double k = 5.0E-07;
//
// Set X values.
//
const double x_min = 0.0;
const double x_max = 0.3;
const int x_num = 11;
const double x_delt = (x_max - x_min) / (x_num - 1);
double[] x = new double[x_num];
for (i = 0; i < x_num; i++)
{
x[i] = ((x_num - i - 1) * x_min
+ i * x_max)
/ (x_num - 1);
}
//
// Set T values.
//
const double t_min = 0.0;
const double t_max = 22000.0;
const int t_num = 51;
const double t_delt = (t_max - t_min) / (t_num - 1);
double[] t = new double[t_num];
for (j = 0; j < t_num; j++)
{
t[j] = ((t_num - j - 1) * t_min
+ j * t_max)
/ (t_num - 1);
}
//
// Set the initial data, for time T_MIN.
//
double[] u = new double[x_num * t_num];
u0(x_min, x_max, t_min, x_num, x, ref u);
//
// The matrix A does not change with time. We can set it once,
// factor it once, and solve repeatedly.
//
double w = k * t_delt / x_delt / x_delt;
double[] a = new double[3 * x_num];
a[0 + 0 * 3] = 0.0;
a[1 + 0 * 3] = 1.0;
a[0 + 1 * 3] = 0.0;
for (i = 1; i < x_num - 1; i++)
{
a[2 + (i - 1) * 3] = -w;
a[1 + i * 3] = 1.0 + 2.0 * w;
a[0 + (i + 1) * 3] = -w;
}
a[2 + (x_num - 2) * 3] = 0.0;
a[1 + (x_num - 1) * 3] = 1.0;
a[2 + (x_num - 1) * 3] = 0.0;
//
// Factor the matrix.
//
typeMethods.r83_np_fa(x_num, ref a);
double[] b = new double[x_num];
double[] fvec = new double[x_num];
for (j = 1; j < t_num; j++)
{
//
// Set the right hand side B.
//
b[0] = ua(x_min, x_max, t_min, t[j]);
f(x_min, x_max, t_min, t[j - 1], x_num, x, ref fvec);
for (i = 1; i < x_num - 1; i++)
{
b[i] = u[i + (j - 1) * x_num] + t_delt * fvec[i];
}
b[x_num - 1] = ub(x_min, x_max, t_min, t[j]);
int job = 0;
fvec = typeMethods.r83_np_sl(x_num, a, b, job);
for (i = 0; i < x_num; i++)
{
u[i + j * x_num] = fvec[i];
}
}
const string x_file = "x.txt";
DTable.dtable_write(x_file, 1, x_num, x, false);
Console.WriteLine("");
Console.WriteLine(" X data written to \"" + x_file + "\".");
const string t_file = "t.txt";
DTable.dtable_write(t_file, 1, t_num, t, false);
Console.WriteLine(" T data written to \"" + t_file + "\".");
const string u_file = "u.txt";
DTable.dtable_write(u_file, x_num, t_num, u, false);
Console.WriteLine(" U data written to \"" + u_file + "\".");
Console.WriteLine("");
Console.WriteLine("FD1D_HEAT_IMPLICIT");
Console.WriteLine(" Normal end of execution.");
Console.WriteLine("");
}