seulex.C

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/*---------------------------------------------------------------------------*\
  =========                 |
  \\      /  F ield         | OpenFOAM: The Open Source CFD Toolbox
   \\    /   O peration     | Website:  https://openfoam.org
    \\  /    A nd           | Copyright (C) 2013-2019 OpenFOAM Foundation
     \\/     M anipulation  |
-------------------------------------------------------------------------------
License
    This file is part of OpenFOAM.

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    under the terms of the GNU General Public License as published by
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    FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
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#include "seulex.H"
#include "addToRunTimeSelectionTable.H"

// * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //

namespace Foam
{
    defineTypeNameAndDebug(seulex, 0);

    addToRunTimeSelectionTable(ODESolver, seulex, dictionary);

    const scalar
        seulex::stepFactor1_ = 0.6,
        seulex::stepFactor2_ = 0.93,
        seulex::stepFactor3_ = 0.1,
        seulex::stepFactor4_ = 4,
        seulex::stepFactor5_ = 0.5,
        seulex::kFactor1_ = 0.7,
        seulex::kFactor2_ = 0.9;
}


// * * * * * * * * * * * * * * * * Constructors  * * * * * * * * * * * * * * //

Foam::seulex::seulex(const ODESystem& ode, const dictionary& dict)
:
    ODESolver(ode, dict),
    jacRedo_(min(1e-4, min(relTol_))),
    nSeq_(iMaxx_),
    cpu_(iMaxx_),
    coeff_(iMaxx_, iMaxx_),
    theta_(2*jacRedo_),
    table_(kMaxx_, n_),
    dfdx_(n_),
    dfdy_(n_),
    a_(n_),
    pivotIndices_(n_),
    dxOpt_(iMaxx_),
    temp_(iMaxx_),
    y0_(n_),
    ySequence_(n_),
    scale_(n_),
    dy_(n_),
    yTemp_(n_),
    dydx_(n_)
{
    // The CPU time factors for the major parts of the algorithm
    const scalar cpuFunc = 1, cpuJac = 5, cpuLU = 1, cpuSolve = 1;

    nSeq_[0] = 2;
    nSeq_[1] = 3;

    for (int i=2; i<iMaxx_; i++)
    {
        nSeq_[i] = 2*nSeq_[i-2];
    }
    cpu_[0] = cpuJac + cpuLU + nSeq_[0]*(cpuFunc + cpuSolve);

    for (int k=0; k<kMaxx_; k++)
    {
        cpu_[k+1] = cpu_[k] + (nSeq_[k+1]-1)*(cpuFunc + cpuSolve) + cpuLU;
    }

    // Set the extrapolation coefficients array
    for (int k=0; k<iMaxx_; k++)
    {
        for (int l=0; l<k; l++)
        {
            scalar ratio = scalar(nSeq_[k])/nSeq_[l];
            coeff_(k, l) = 1/(ratio - 1);
        }
    }
}


// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * //

bool Foam::seulex::seul
(
    const scalar x0,
    const scalarField& y0,
    const label li,
    const scalar dxTot,
    const label k,
    scalarField& y,
    const scalarField& scale
) const
{
    label nSteps = nSeq_[k];
    scalar dx = dxTot/nSteps;

    for (label i=0; i<n_; i++)
    {
        for (label j=0; j<n_; j++)
        {
            a_(i, j) = -dfdy_(i, j);
        }

        a_(i, i) += 1/dx;
    }

    LUDecompose(a_, pivotIndices_);

    scalar xnew = x0 + dx;
    odes_.derivatives(xnew, y0, li, dy_);
    LUBacksubstitute(a_, pivotIndices_, dy_);

    yTemp_ = y0;

    for (label nn=1; nn<nSteps; nn++)
    {
        yTemp_ += dy_;
        xnew += dx;

        if (nn == 1 && k<=1)
        {
            scalar dy1 = 0;
            for (label i=0; i<n_; i++)
            {
                dy1 += sqr(dy_[i]/scale[i]);
            }
            dy1 = sqrt(dy1);

            odes_.derivatives(x0 + dx, yTemp_, li, dydx_);
            for (label i=0; i<n_; i++)
            {
                dy_[i] = dydx_[i] - dy_[i]/dx;
            }

            LUBacksubstitute(a_, pivotIndices_, dy_);

            // This form from the original paper is unreliable
            // step size underflow for some cases
            // const scalar denom = max(1, dy1);

            // This form is reliable but limits how large the step size can be
            const scalar denom = min(1, dy1 + small);

            scalar dy2 = 0;
            for (label i=0; i<n_; i++)
            {
                // Test of dy_[i] to avoid overflow
                if (mag(dy_[i]) > scale[i]*denom)
                {
                    theta_ = 1;
                    return false;
                }

                dy2 += sqr(dy_[i]/scale[i]);
            }
            dy2 = sqrt(dy2);
            theta_ = dy2/denom;

            if (theta_ > 1)
            {
                return false;
            }
        }

        odes_.derivatives(xnew, yTemp_, li, dy_);
        LUBacksubstitute(a_, pivotIndices_, dy_);
    }

    for (label i=0; i<n_; i++)
    {
        y[i] = yTemp_[i] + dy_[i];
    }

    return true;
}


void Foam::seulex::extrapolate
(
    const label k,
    scalarRectangularMatrix& table,
    scalarField& y
) const
{
    for (int j=k-1; j>0; j--)
    {
        for (label i=0; i<n_; i++)
        {
            table[j-1][i] =
                table(j, i) + coeff_(k, j)*(table(j, i) - table[j-1][i]);
        }
    }

    for (int i=0; i<n_; i++)
    {
        y[i] = table(0, i) + coeff_(k, 0)*(table(0, i) - y[i]);
    }
}


bool Foam::seulex::resize()
{
    if (ODESolver::resize())
    {
        table_.shallowResize(kMaxx_, n_);
        resizeField(dfdx_);
        resizeMatrix(dfdy_);
        resizeMatrix(a_);
        resizeField(pivotIndices_);
        resizeField(y0_);
        resizeField(ySequence_);
        resizeField(scale_);
        resizeField(dy_);
        resizeField(yTemp_);
        resizeField(dydx_);

        return true;
    }
    else
    {
        return false;
    }
}


void Foam::seulex::solve
(
    scalar& x,
    scalarField& y,
    const label li,
    stepState& step
) const
{
    temp_[0] = great;
    scalar dx = step.dxTry;
    y0_ = y;
    dxOpt_[0] = mag(0.1*dx);

    if (step.first || step.prevReject)
    {
        theta_ = 2*jacRedo_;
    }

    if (step.first)
    {
        // NOTE: the first element of relTol_ and absTol_ are used here.
        scalar logTol = -log10(relTol_[0] + absTol_[0])*0.6 + 0.5;
        kTarg_ = max(1, min(kMaxx_ - 1, int(logTol)));
    }

    forAll(scale_, i)
    {
        scale_[i] = absTol_[i] + relTol_[i]*mag(y[i]);
    }

    bool jacUpdated = false;

    if (theta_ > jacRedo_)
    {
        odes_.jacobian(x, y, li, dfdx_, dfdy_);
        jacUpdated = true;
    }

    int k;
    scalar dxNew = mag(dx);
    bool firstk = true;

    while (firstk || step.reject)
    {
        dx = step.forward ? dxNew : -dxNew;
        firstk = false;
        step.reject = false;

        if (mag(dx) <= mag(x)*sqr(small))
        {
             WarningInFunction
                 << "step size underflow :"  << dx << endl;
        }

        scalar errOld = 0;

        for (k=0; k<=kTarg_+1; k++)
        {
            bool success = seul(x, y0_, li, dx, k, ySequence_, scale_);

            if (!success)
            {
                step.reject = true;
                dxNew = mag(dx)*stepFactor5_;
                break;
            }

            if (k == 0)
            {
                 y = ySequence_;
            }
            else
            {
                forAll(ySequence_, i)
                {
                    table_[k-1][i] = ySequence_[i];
                }
            }

            if (k != 0)
            {
                extrapolate(k, table_, y);
                scalar err = 0;
                forAll(scale_, i)
                {
                    scale_[i] = absTol_[i] + relTol_[i]*mag(y0_[i]);
                    err += sqr((y[i] - table_(0, i))/scale_[i]);
                }
                err = sqrt(err/n_);
                if (err > 1/small || (k > 1 && err >= errOld))
                {
                    step.reject = true;
                    dxNew = mag(dx)*stepFactor5_;
                    break;
                }
                errOld = min(4*err, 1);
                scalar expo = 1.0/(k + 1);
                scalar facmin = pow(stepFactor3_, expo);
                scalar fac;
                if (err == 0)
                {
                    fac = 1/facmin;
                }
                else
                {
                    fac = stepFactor2_/pow(err/stepFactor1_, expo);
                    fac = max(facmin/stepFactor4_, min(1/facmin, fac));
                }
                dxOpt_[k] = mag(dx*fac);
                temp_[k] = cpu_[k]/dxOpt_[k];

                if ((step.first || step.last) && err <= 1)
                {
                    break;
                }

                if
                (
                    k == kTarg_ - 1
                 && !step.prevReject
                 && !step.first && !step.last
                )
                {
                    if (err <= 1)
                    {
                        break;
                    }
                    else if (err > nSeq_[kTarg_]*nSeq_[kTarg_ + 1]*4)
                    {
                        step.reject = true;
                        kTarg_ = k;
                        if (kTarg_>1 && temp_[k-1] < kFactor1_*temp_[k])
                        {
                            kTarg_--;
                        }
                        dxNew = dxOpt_[kTarg_];
                        break;
                    }
                }

                if (k == kTarg_)
                {
                    if (err <= 1)
                    {
                        break;
                    }
                    else if (err > nSeq_[k + 1]*2)
                    {
                        step.reject = true;
                        if (kTarg_>1 && temp_[k-1] < kFactor1_*temp_[k])
                        {
                            kTarg_--;
                        }
                        dxNew = dxOpt_[kTarg_];
                        break;
                    }
                }

                if (k == kTarg_+1)
                {
                    if (err > 1)
                    {
                        step.reject = true;
                        if
                        (
                            kTarg_ > 1
                         && temp_[kTarg_-1] < kFactor1_*temp_[kTarg_]
                        )
                        {
                            kTarg_--;
                        }
                        dxNew = dxOpt_[kTarg_];
                    }
                    break;
                }
            }
        }
        if (step.reject)
        {
            step.prevReject = true;
            if (!jacUpdated)
            {
                theta_ = 2*jacRedo_;

                if (theta_ > jacRedo_ && !jacUpdated)
                {
                    odes_.jacobian(x, y, li, dfdx_, dfdy_);
                    jacUpdated = true;
                }
            }
        }
    }

    jacUpdated = false;

    step.dxDid = dx;
    x += dx;

    label kopt;
    if (k == 1)
    {
        kopt = 2;
    }
    else if (k <= kTarg_)
    {
        kopt=k;
        if (temp_[k-1] < kFactor1_*temp_[k])
        {
            kopt = k - 1;
        }
        else if (temp_[k] < kFactor2_*temp_[k - 1])
        {
            kopt = min(k + 1, kMaxx_ - 1);
        }
    }
    else
    {
        kopt = k - 1;
        if (k > 2 && temp_[k-2] < kFactor1_*temp_[k - 1])
        {
            kopt = k - 2;
        }
        if (temp_[k] < kFactor2_*temp_[kopt])
        {
            kopt = min(k, kMaxx_ - 1);
        }
    }

    if (step.prevReject)
    {
        kTarg_ = min(kopt, k);
        dxNew = min(mag(dx), dxOpt_[kTarg_]);
        step.prevReject = false;
    }
    else
    {
        if (kopt <= k)
        {
            dxNew = dxOpt_[kopt];
        }
        else
        {
            if (k < kTarg_ && temp_[k] < kFactor2_*temp_[k - 1])
            {
                dxNew = dxOpt_[k]*cpu_[kopt + 1]/cpu_[k];
            }
            else
            {
                dxNew = dxOpt_[k]*cpu_[kopt]/cpu_[k];
            }
        }
        kTarg_ = kopt;
    }

    step.dxTry = step.forward ? dxNew : -dxNew;
}


// ************************************************************************* //