/// <summary>
/// Returns a Backward-Differentiation-Formula of order <paramref name="i"/>.
/// </summary>
static public BDFSchemeCoeffs BDF(int i)
{
BDFSchemeCoeffs R = new BDFSchemeCoeffs();
switch (i)
{
case 1:
R.theta1 = 1.0;
R.theta0 = 0.0;
R.beta = new[] { 1.0 };
break;
case 2:
R.theta1 = 2.0 / 3.0;
R.theta0 = 0.0;
R.beta = new[] { 4.0 / 3.0, -1.0 / 3.0 };
break;
case 3:
R.theta1 = 6.0 / 11.0;
R.theta0 = 0.0;
R.beta = new[] { 18.0 / 11.0, -9.0 / 11.0, 2.0 / 11.0 };
break;
case 4:
R.theta1 = 12.0 / 25.0;
R.theta0 = 0.0;
R.beta = new[] { 48.0 / 25.0, -36.0 / 25.0, 16.0 / 25, -3.0 / 25.0 };
break;
case 5:
R.theta1 = 60.0 / 137.0;
R.theta0 = 0.0;
R.beta = new[] { 300.0 / 137.0, -300.0 / 137.0, 200.0 / 137.0, -75.0 / 137.0, 12.0 / 137.0 };
break;
case 6:
R.theta1 = 60.0 / 147.0;
R.theta0 = 0.0;
R.beta = new[] { 360.0 / 147.0, -450.0 / 147.0, 400.0 / 147.0, -225.0 / 147.0, 72.0 / 147.0, -10.0 / 147.0 };
break;
default:
throw new NotImplementedException("BDF scheme of order " + i + " is not supported.");
}
if (Math.Abs(R.beta.Sum() - 1.0) > 1.0e-10)
{
throw new ApplicationException();
}
return(R);
}
/// <summary>
/// Returns an Explicit-Euler formula.
/// </summary>
static public BDFSchemeCoeffs ExplicitEuler()
{
BDFSchemeCoeffs R = new BDFSchemeCoeffs();
R.theta1 = 0.0;
R.theta0 = 1.0;
R.beta = new[] { 1.0 };
if (Math.Abs(R.beta.Sum() - 1.0) > 1.0e-10)
{
throw new ApplicationException();
}
return(R);
}
/// <summary>
/// Returns a Crank-Nicolson formula.
/// </summary>
static public BDFSchemeCoeffs CrankNicolson()
{
BDFSchemeCoeffs R = new BDFSchemeCoeffs();
R.theta1 = 0.5;
R.theta0 = 0.5;
R.beta = new[] { 1.0 };
if (Math.Abs(R.beta.Sum() - 1.0) > 1.0e-10)
{
throw new ApplicationException();
}
return(R);
}
/// <summary>
/// Initilizes a Chain of BDF-Schemes starting from ImplicitEuler up to the desired order,
/// this is required e.g. when starting a simulation.
/// Then the first step is done by implicit Euler, the second step by BDF2 and so forth
/// </summary>
/// <param name="BDForder"></param>
/// <returns>An Array of BDSchemes from implicit euler, entry[i] up to the desired order, entry[0]
/// Hack:
/// 0 = Explicit Euler
/// -1 = Crank-Nicholson
/// </returns>
public static BDFSchemeCoeffs[] GetChain(int BDForder)
{
BDFSchemeCoeffs[] SchemeChain;
switch (BDForder)
{
case -1:
SchemeChain = new BDFSchemeCoeffs[] { BDFSchemeCoeffs.CrankNicolson() };
break;
case 0:
SchemeChain = new BDFSchemeCoeffs[] { BDFSchemeCoeffs.ExplicitEuler() };
break;
default:
SchemeChain = new BDFSchemeCoeffs[BDForder];
for (int i = BDForder; i >= 1; i--)
{
SchemeChain[BDForder - i] = BDFSchemeCoeffs.BDF(i);
}
break;
}
return(SchemeChain);
}
