我试图用Thomas算法解决差热方程。
身体问题：我们有插头，左侧有温度0，右侧温度为1。
对于Thomas算法，我编写了一个函数，它接受三个QVector和int个数量的方程。
这是我的代码：
#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;
}
我的问题是： 当我编译并运行它时，我得到了这个：
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
一次又一次地循环。
我不知道为什么我得到这个。
也许我的错误在于我的托马斯方法功能的分配Tn结果？
或实现托马斯方法？或在边界条件下？
答案 0 :(得分：0)
我明白了！
边界条件必须作用于矢量
main_diagonal.first()=1;
main_diagonal.last()=1;
beyond_main_diagonal.first()=0;
below_main_diagonal.last()=0;
因此T [0]必须为0且T [N-1]必须为1.我们可以这样做：
for(int i(0);i<N;++i){
main_diagonal[i]*=(-1);
}
由于这个T [0]将始终等于零而T [N-1]将等于1;
在我读到关于Thomas方法的文章中，第一步是否定主对角线，我已经完成了，但是在函数结束时我必须做反向的事情，所以：
#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;
}
我们可以再次使用这个功能，这不是绝对最佳的，但它工作稳定。
然后，整个代码将如下所示：
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
并且在最后一步中我们将获得绝对的“物理”结果：
{{1}}