Solution of heat equation by Thomas algorithm

Question

I am trying to solve differential heat equation using Thomas algorithm.

Physical problem : We have plug, left side is having temperature 0 , right side temperature is 1 .

For Thomas algorithm I have written a function, which accept three QVector and int value amount of equations.

This is my code:

#include <QCoreApplication>
#include <QVector>
#include <QDebug>
#include <iostream>
using std::cin;

void enterIn(QVector<float> &Array, int Amount_of_elements){
    int transit;
    for(int i=0;i<Amount_of_elements;i++){
        cin>>transit;
        Array.push_back(transit);
    }
}

QVector<float> shuttle_method(const QVector<float> &below_main_diagonal,
                              QVector<float> &main_diagonal,
                              const QVector<float> &beyond_main_diagonal,
                              const QVector<float> &free_term,
                              const int N){
    QVector <float> c;
    QVector <float> d;

    for(int i=0;i<N;i++){
        main_diagonal[i]*=(-1);
    }

    QVector<float> x; //result

    c.push_back(beyond_main_diagonal[0]/main_diagonal[0]);
    d.push_back(-free_term[0]/main_diagonal[0]);

    for(int i=1;i<=N-2;i++){
        c.push_back(beyond_main_diagonal[i]/(main_diagonal[i]-below_main_diagonal[i]*c[i-1]));
        d.push_back(  (below_main_diagonal[i]*d[i-1]  -  free_term[i])  /  (main_diagonal[i]-  below_main_diagonal[i]*c[i-1])  );
    }

    x.resize(N);
    //qDebug()<<x.size()<<endl;

    int n=N-1;
    x[n]=(below_main_diagonal[n]*d[n-1]-free_term[n])/(main_diagonal[n]-below_main_diagonal[n]*c[n-1]);

    for(int i=n-1;i>=0;i--){
        x[i]=c[i]*x[i+1]+d[i];
        // qDebug()<<x[i]<<endl;
    }

    return x;
}

int main()
{    
    QVector <float> alpha;  // below
    QVector <float> beta;   // main diagonal * (-1)
    QVector <float> gamma;  // beyond
    QVector <float> b;      // free term

    QVector<float> T;

    int cells_x=40;         //amount of equations
    alpha.resize(cells_x);
    beta.resize(cells_x);
    gamma.resize(cells_x);
    b.resize(cells_x);
    T.resize(cells_x);

    float dt=0.2,h=0.1;

    alpha[0]=0;
    for(int i=1;i<cells_x;i++){
        alpha[i]= -dt/(h*h);
    }

    for(int i=0;i<cells_x;i++){
        beta[i] = (2*dt)/(h*h)+1;
    }
    for(int i=0;i<cells_x-1;i++){
        gamma[i]= -dt/(h*h);
    }
    gamma[cells_x-1]=0;

    qDebug()<<"alpha= "<<endl<<alpha.size()<<alpha<<endl<<"beta = "<<endl<<beta.size()<<beta<<endl<<"gamma= "<<gamma.size()<<gamma<<endl;


    for(int i=0;i<cells_x-1;i++){
        T[i]=0;
    }
    T[cells_x-1]=1;
    qDebug()<<endl<<endl<<T<<endl;

    //qDebug()<< shuttle_method(alpha,beta,gamma,b,N);

    QVector<float> Tn;
    Tn.resize(cells_x);
    Tn = shuttle_method(alpha,beta,gamma,T,cells_x);
    Tn[0]=0;Tn[cells_x-1]=1;
    for(int stepTime = 0; stepTime < 50; stepTime++){
        Tn = shuttle_method(alpha,beta,gamma,Tn,cells_x);
        Tn[0]=0;
        Tn[cells_x-1]=1;
        qDebug()<<Tn<<endl;
    }
    return 0;
}

My problem is: when I compile and run it I am getting this:

Tn  <20 items>  QVector<float>
0   float
0.000425464 float
0.000664658 float
0.000937085 float
0.00125637  float
0.00163846  float
0.00210249  float
0.00267163  float
0.00337436  float
0.00424581  float
0.00532955  float
0.00667976  float
0.00836396  float
0.0104664   float
0.0130921   float
0.0163724   float
0.0204714   float
0.0255939   float
0.0319961   float

Tn  <20 items>  QVector<float>
0   float
-0.000425464    float
0.000643385 float
-0.000926707    float
0.00120951  float
-0.00161561 float
0.00202056  float
-0.00263167 float
0.00324078  float
-0.00418065 float
0.00511726  float
-0.00657621 float
0.00802998  float
-0.0103034  float
0.0125688   float
-0.0161171  float
0.0196527   float
-0.0251945  float
0.0307164   float
1   float

Tn  <20 items>  QVector<float>
0   float
0.000425464 float
0.000664658 float
0.000937085 float
0.00125637  float
0.00163846  float
0.00210249  float
0.00267163  float
0.00337436  float
0.00424581  float
0.00532955  float
0.00667976  float
0.00836396  float
0.0104664   float
0.0130921   float
0.0163724   float
0.0204714   float
0.0255939   float
0.0319961   float

Tn  <20 items>  QVector<float>
0   float
-0.000425464    float
0.000643385 float
-0.000926707    float
0.00120951  float
-0.00161561 float
0.00202056  float
-0.00263167 float
0.00324078  float
-0.00418065 float
0.00511726  float
-0.00657621 float
0.00802998  float
-0.0103034  float
0.0125688   float
-0.0161171  float
0.0196527   float
-0.0251945  float
0.0307164   float
1   float

Again and again in loop.
I have no idea why I am getting this.

Maybe my mistake is in assign Tn result of my Thomas-method-function?
or in realization of Thomas method? or in boundary conditions?

Answer 1

I got it!

Boundary conditions must be acting to vectors

QVector<float> below_main_diagonal, 
QVector<float>  main_diagonal, 
QVector<float>  beyond_main_diagonal

so that T[0] must be 0 and T[N-1] must be 1. We can do it this way:

main_diagonal.first()=1;
main_diagonal.last()=1;
beyond_main_diagonal.first()=0;
below_main_diagonal.last()=0;

and due to this T[0] will be always equal to zero and T[N-1] will be equal to 1;

And in the article where I read about Thomas method the first step was to negate main diagonal, I have done it, but then in the end of the function I must do reverse thing, so:

for(int i(0);i<N;++i){
   main_diagonal[i]*=(-1);
}

and we can use this function again, this is not absolutely optimal, but it is working stable.

Then, the whole code will be look like this:

#include <QCoreApplication>
#include <QVector>
#include <QDebug>
#include <iostream>


QVector<float> Thomas_Algorithm( QVector<float> &below_main_diagonal ,
                                     QVector<float> &main_diagonal  ,
                                     QVector<float> &beyond_main_diagonal ,
                                     QVector<float> &free_term,
                                     const  int N){

        QVector<float> x; //vector of result

        // checking of input data
        if(below_main_diagonal.size()!=main_diagonal.size()||
                main_diagonal.size()!=beyond_main_diagonal.size()||
                free_term.size()!=main_diagonal.size())
        { qDebug()<<"Error!\n"
                    "Error with accepting Arrays! Dimensities are different!"<<endl;
            x.resize(0);
            return x;
        }
        if(below_main_diagonal[0]!=0){
            qDebug()<< "Error!\n"
                       "First element of below_main_diagonal must be equal to zero!"<<endl;
            x.resize(0);
            return x;
        }
        if(beyond_main_diagonal.last()!=0){
            qDebug()<< "Error!\n"
                       "Last element of beyond_main_diagonal must be equal to zero!"<<endl;
            x.resize(0);
            return x;

        }
        // end of checking

        QVector <float> c;
        QVector <float> d;

        for(int i=0;i<N;i++){
            main_diagonal[i]*=(-1);
        }

        c.push_back(beyond_main_diagonal[0]/main_diagonal[0]);
        d.push_back(-free_term[0]/main_diagonal[0]);

        for(int i=1;i<=N-2;i++){
            c.push_back(beyond_main_diagonal[i]/(main_diagonal[i]-below_main_diagonal[i]*c[i-1]));
            d.push_back(  (below_main_diagonal[i]*d[i-1]  -  free_term[i])  /
                    (main_diagonal[i]-  below_main_diagonal[i]*c[i-1])  );
        }

        x.resize(N);

        int n=N-1;
        x[n]=(below_main_diagonal[n]*d[n-1]-free_term[n])/(main_diagonal[n]-below_main_diagonal[n]*c[n-1]);

        for(int i=n-1;i>=0;i--){
            x[i]=c[i]*x[i+1]+d[i];
        }

        for(int i(0);i<N;++i){
           main_diagonal[i]*=(-1);
        }

        return x;
    }

    int main()
    { 
        QVector <float> alpha;  // below
        QVector <float> beta;   // main diagonal * (-1)
        QVector <float> gamma;  // beyond
        QVector <float> b;      // free term


        QVector<float> T;

        int cells_x=30;         // amount of steps
        alpha.resize(cells_x);
        beta.resize(cells_x);
        gamma.resize(cells_x);
        T.resize(cells_x );

        float dt=0.2,h=0.1;

        alpha[0]=0;
        for(int i=1;i<cells_x-1;i++){
            alpha[i]= -dt/(h*h);
        }
        alpha[cells_x-1]=0;

        beta[0]=1;
        for(int i=1;i<cells_x-1;i++){
            beta[i] = (2*dt)/(h*h)+1;
        }
        beta[cells_x-1]=1;
        gamma[0]=0;
        for(int i=1;i<cells_x-1;i++){
            gamma[i]= -dt/(h*h);
        }
        gamma[cells_x-1]=0;  

        for(int i=0;i<cells_x-1;i++){
            T[i]=0;
        }
        T[cells_x-1]=1;

        QVector<float>Tn;
        Tn.resize(cells_x);
        Tn=  Thomas_Algorithm(alpha,beta,gamma,T,cells_x);
        // boundary conditions!
        beta.first()=1;
        beta.last()=1;
        gamma.first()=0;
        alpha.last()=0;
        // and then due to bc we always have T[0]=0 and T[n]=1

        for(int stepTime=0;stepTime<100;stepTime++){
            Tn = Thomas_Algorithm(alpha,beta,gamma,Tn,cells_x);

            qDebug()<<"stepTime = "<<stepTime<<endl<<endl;
            qDebug()<<Tn<<endl;

            // boundary conditions!
            beta.first()=1;
            beta.last()=1;
            gamma.first()=0;
            alpha.last()=0;
            // and then due to bc we always have T[0]=0 and T[n]=1
        }

        return 0;
    }

and in the last step we are going to get absolutely "physical" results:

Tn  <30 items>  QVector<float>
                0   float
                0.0344828   float
                0.0689656   float
                0.103448    float
                0.137931    float
                0.172414    float
                0.206897    float
                0.24138 float
                0.275862    float
                0.310345    float
                0.344828    float
                0.379311    float
                0.413793    float
                0.448276    float
                0.482759    float
                0.517242    float
                0.551724    float
                0.586207    float
                0.62069 float
                0.655173    float
                0.689655    float
                0.724138    float
                0.758621    float
                0.793104    float
                0.827586    float
                0.862069    float
                0.896552    float
                0.931035    float
                0.965517    float
                1   float