numericdifferentiation.hh

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// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
// vi: set et ts=4 sw=4 sts=4:
//
// SPDX-FileCopyrightText: Copyright © DuMux Project contributors, see AUTHORS.md in root folder
// SPDX-License-Identifier: GPL-3.0-or-later
//
#ifndef DUMUX_NUMERIC_DIFFERENTIATION_HH
#define DUMUX_NUMERIC_DIFFERENTIATION_HH
 
#include <cmath>
#include <cassert>
#include <limits>
 
#include <dune/common/ftraits.hh>
 
namespace Dumux {

class NumericDifferentiation
{
public:

    template<class Scalar, class Magnitude = typename Dune::FieldTraits<Scalar>::real_type>
    static Magnitude epsilon(const Scalar value, const Magnitude baseEps = 1e-10)
    {
        assert(std::numeric_limits<Magnitude>::epsilon()*1e4 < baseEps);
        // the epsilon value used for the numeric differentiation is
        // now scaled by the absolute value of the primary variable...
        using std::abs;
        return baseEps*(abs(value) + 1.0);
    }

    template<class Function, class Scalar, class FunctionEvalType>
    static void partialDerivative(const Function& function, Scalar x0,
                                  FunctionEvalType& derivative,
                                  const FunctionEvalType& fx0,
                                  const int numericDifferenceMethod = 1)
    { partialDerivative(function, x0, derivative, fx0, epsilon(x0), numericDifferenceMethod); }

    template<class Function, class Scalar, class EpsType, class FunctionEvalType>
    static void partialDerivative(const Function& function, Scalar x0,
                                  FunctionEvalType& derivative,
                                  const FunctionEvalType& fx0,
                                  const EpsType eps,
                                  const int numericDifferenceMethod = 1)
    requires(
        std::is_same_v<Scalar, decltype(std::declval<Scalar>() + std::declval<EpsType>())>
        && std::is_same_v<Scalar, decltype(std::declval<Scalar>() - std::declval<EpsType>())>
    ){
        // Five-point stencil numeric difference,
        // Abramowitz & Stegun, Table 25.2.
        // The error is proportional to eps^4.
        if (numericDifferenceMethod == 5)
        {
            derivative = function(x0 + eps);
            derivative -= function(x0 - eps);
            derivative *= 8.0;
            derivative += function(x0 - 2.0*eps);
            derivative -= function(x0 + 2.0*eps);
            derivative /= 12.0*eps;
            return;
        }
 
        // Forward, central, or backward differences
        EpsType delta = 0.0;
 
        // we are using forward or central differences, i.e. we need to calculate f(x + \epsilon)
        if (numericDifferenceMethod >= 0)
        {
            delta += eps;
            // calculate the function evaluated with the deflected variable
            derivative = function(x0 + eps);
        }
 
        // we are using backward differences,
        // i.e. we don't need to calculate f(x + \epsilon)
        // we can recycle the (possibly cached) f(x)
        else derivative = fx0;
 
        // we are using backward or central differences,
        // i.e. we need to calculate f(x - \epsilon)
        if (numericDifferenceMethod <= 0)
        {
            delta += eps;
            // subtract the function evaluated with the deflected variable
            derivative -= function(x0 - eps);
        }
 
        // we are using forward differences,
        // i.e. we don't need to calculate f(x - \epsilon)
        // we can recycle the (possibly cached) f(x)
        else derivative -= fx0;
 
        // divide difference in residuals by the magnitude of the
        // deflections between the two function evaluation
        derivative /= delta;
    }
};
 
} // end namespace Dumux
 
#endif