对于不同的微分矩阵，热方程计算变得不稳定
我想使用RK4算法在一个维度上求解热方程。现在我可以将热方程写为
d_t u = d_x(A d_x u)
或
d_t u = A d_x^2(u)
A一个常数。
我想比较两种方法，因此编写了以下代码：
#include <iostream>
#include <math.h>
#include <armadillo>
#define Y0 0
#define Y1 1
#define YSIZE 400
#define ALPHA 1
auto rk4(double f(double, double))
{
return
[ f ](double t, double y, double dt ) -> double{ return
[t,y,dt,f ]( double dy1) -> double{ return
[t,y,dt,f,dy1 ]( double dy2) -> double{ return
[t,y,dt,f,dy1,dy2 ]( double dy3) -> double{ return
[t,y,dt,f,dy1,dy2,dy3]( double dy4) -> double{ return
( dy1 + 2*dy2 + 2*dy3 + dy4 ) / 6 ;} (
dt * f( t+dt , y+dy3 ) );} (
dt * f( t+dt/2, y+dy2/2 ) );} (
dt * f( t+dt/2, y+dy1/2 ) );} (
dt * f( t , y ) );} ;
}
auto rk4(arma::colvec f(double, arma::colvec))
{
return
[ f ](double t, arma::colvec y, double dt ) -> arma::colvec{ return
[t,y,dt,f ]( arma::colvec dy1) -> arma::colvec{ return
[t,y,dt,f,dy1 ]( arma::colvec dy2) -> arma::colvec{ return
[t,y,dt,f,dy1,dy2 ]( arma::colvec dy3) -> arma::colvec{ return
[t,y,dt,f,dy1,dy2,dy3]( arma::colvec dy4) -> arma::colvec{ return
( dy1 + 2*dy2 + 2*dy3 + dy4 ) / 6 ;} (
dt * f( t+dt , y+dy3 ) );} (
dt * f( t+dt/2, y+dy2/2 ) );} (
dt * f( t+dt/2, y+dy1/2 ) );} (
dt * f( t , y ) );} ;
}
void create_first_derivative_matrix(const size_t size, arma::mat &matrix, double dh = -1)
{
if(dh <= 0)
dh = ((double)Y1 - (double)Y0) / ((double)YSIZE - 1.);
matrix = arma::mat(size, size, arma::fill::zeros);
matrix.diag(1) = arma::colvec(size - 1, arma::fill::ones) * 8;
matrix.diag(-1) = arma::colvec(size - 1, arma::fill::ones) * -8;
matrix.diag(2) = arma::colvec(size - 2, arma::fill::ones) * -1;
matrix.diag(-2) = arma::colvec(size - 2, arma::fill::ones);
#ifdef USE_REFLECT
matrix(0, 0) = 0;
matrix(0, 1) = 0;
matrix(0, 2) = 0;
matrix(0, 3) = 0;
matrix(0, 4) = 0;
matrix(1, 0) = -8;
matrix(1, 1) = 1;
matrix(1, 2) = 8;
matrix(1, 3) = -1;
matrix(1, 4) = 0;
matrix(size - 1, size - 1) = 0;
matrix(size - 1, size - 2) = 0;
matrix(size - 1, size - 3) = 0;
matrix(size - 1, size - 4) = 0;
matrix(size - 1, size - 5) = 0;
matrix(size - 2, size - 1) = 8;
matrix(size - 2, size - 2) = -1;
matrix(size - 2, size - 3) = -8;
matrix(size - 2, size - 4) = 1;
matrix(size - 2, size - 5) = 0;
#else
matrix(0, 0) = -25;
matrix(0, 1) = 48;
matrix(0, 2) = -36;
matrix(0, 3) = 16;
matrix(0, 4) = -3;
matrix(1, 0) = -3;
matrix(1, 1) = -10;
matrix(1, 2) = 18;
matrix(1, 3) = -6;
matrix(1, 4) = 1;
matrix(size - 1, size - 1) = 25;
matrix(size - 1, size - 2) = -48;
matrix(size - 1, size - 3) = 36;
matrix(size - 1, size - 4) = -16;
matrix(size - 1, size - 5) = 3;
matrix(size - 2, size - 1) = 3;
matrix(size - 2, size - 2) = 10;
matrix(size - 2, size - 3) = -18;
matrix(size - 2, size - 4) = 6;
matrix(size - 2, size - 5) = -1;
#endif
matrix = matrix / (12 * dh);
}
void create_second_derivative_matrix(const size_t size, arma::mat &matrix, double dh = -1)
{
if(dh <= 0)
dh = ((double)Y1 - (double)Y0) / ((double)YSIZE - 1.);
matrix = arma::mat(size, size, arma::fill::zeros);
matrix.diag(1) = arma::colvec(size - 1, arma::fill::ones) * 16;
matrix.diag(-1) = arma::colvec(size - 1, arma::fill::ones) * 16;
matrix.diag(2) = arma::colvec(size - 2, arma::fill::ones) * -1;
matrix.diag(-2) = arma::colvec(size - 2, arma::fill::ones) * -1;
matrix.diag(0) = arma::colvec(size, arma::fill::ones) * -30;
matrix(0, 0) = 35;
matrix(0, 1) = -104;
matrix(0, 2) = 114;
matrix(0, 3) = -56;
matrix(0, 4) = 11;
matrix(1, 0) = 11;
matrix(1, 1) = -20;
matrix(1, 2) = 6;
matrix(1, 3) = 4;
matrix(1, 4) = -1;
matrix(size - 1, size - 1) = 35;
matrix(size - 1, size - 2) = -104;
matrix(size - 1, size - 3) = 114;
matrix(size - 1, size - 4) = -56;
matrix(size - 1, size - 5) = 11;
matrix(size - 2, size - 1) = 11;
matrix(size - 2, size - 2) = -20;
matrix(size - 2, size - 3) = 6;
matrix(size - 2, size - 4) = 4;
matrix(size - 2, size - 5) = -1;
matrix = matrix / (12 * dh * dh);
}
arma::colvec eval_heat_eqn(const double t, const arma::colvec y)
{
(void) t;
arma::mat deriv_matrix;
#define USE_FIRST_ORDER
#ifdef USE_FIRST_ORDER
create_first_derivative_matrix(y.size(), deriv_matrix);
#else
create_second_derivative_matrix(y.size(), deriv_matrix);
#endif
arma::colvec heat_vec = arma::colvec(y.size(), arma::fill::zeros);
arma::colvec alpha_vec = arma::colvec(y.size(), arma::fill::ones) * ALPHA;
arma::mat alpha_mat = arma::mat(y.size(), y.size(), arma::fill::zeros);
alpha_mat.diag() = alpha_vec;
for(size_t i = 0; i < y.size(); ++i)
if(i >= 0.5 * y.size())
heat_vec(i) = 0.01;
else
heat_vec(i) = 0;
#ifdef USE_FIRST_ORDER
return alpha_mat * ((deriv_matrix * deriv_matrix) * y) + heat_vec;
#else
return alpha_mat * (deriv_matrix * y) + heat_vec;
#endif
}
//Prints a vector
void print_vector(const arma::colvec &x, const arma::colvec &vector_data, std::string filename)
{
std::ofstream out(filename.c_str());
for(size_t i = 0; i < vector_data.size(); ++i)
out << x(i) << '\t' << abs(vector_data[i]) << '\n';
out.close();
}
int main(void)
{
const double TIME_MAXIMUM = 10, WHOLE_TOLERANCE = 1e-12 ;
const double T_START = 0.0;
double DT = 0.00010;
arma::mat local_deriv_matrix_I, local_deriv_matrix_II;
auto is_whole = [WHOLE_TOLERANCE](double t )->bool { return fabs(t-round(t)) < WHOLE_TOLERANCE; } ;
double t = T_START ;
const double stability_value = ALPHA /(pow(((double)Y1 - (double)Y0) / ((double)YSIZE + 1.), 2));
DT = 0.15 / stability_value;
std::cout << "Stability: " << ALPHA * DT/(pow(((double)Y1 - (double)Y0) / ((double)YSIZE + 1.), 2)) << '\n';
std::cout << "DT: " << DT << '\n';
auto dy_vec = rk4( eval_heat_eqn ) ;
arma::colvec y_vec = arma::colvec(YSIZE, arma::fill::ones);//
arma::colvec test_vec = arma::linspace(Y0, Y1, YSIZE);
arma::colvec x_vec = arma::linspace(Y0, Y1, y_vec.size());
test_vec = arma::pow(test_vec, 3);
create_first_derivative_matrix(y_vec.size(), local_deriv_matrix_I);
create_second_derivative_matrix(y_vec.size(), local_deriv_matrix_II);
arma::colvec deriv_vec = local_deriv_matrix_I * test_vec;
print_vector(x_vec, test_vec, "test_vector.txt");
print_vector(x_vec, deriv_vec, "derivation_vector.txt");
deriv_vec = local_deriv_matrix_I * local_deriv_matrix_I * test_vec;
print_vector(x_vec, deriv_vec, "derivation_II_vector.txt");
deriv_vec = local_deriv_matrix_II * test_vec;
print_vector(x_vec, deriv_vec, "II_derivation_vector.txt");
t = T_START ;
std::vector<arma::colvec> results;
while(t <= TIME_MAXIMUM) {
if (is_whole(t)) { printf("y(%4.2f)[0]\t=(%12.6f)\n", t, y_vec[0]); }
y_vec += dy_vec(t,y_vec,DT) ; t += DT; results.push_back(y_vec);
}
arma::mat result_matrix = arma::mat(y_vec.size(), results.size(), arma::fill::zeros);
for(size_t i = 0; i < results.size(); ++i)
if(i % 100 == 0)
result_matrix.col(i) = results[i];
result_matrix.save("heat_results.txt", arma::raw_ascii);
return 0;
}
现在，在为#define USE_FIRST_ORDER运行此程序时（即使用第一种方法），我的结果看起来像
而当直接使用二阶矩阵时，我得到另一个结果（与第一个版本相比，这是正确的）：
为了进行调试，我将矩阵M1（第一顺序）和M2（第二顺序）与向量相乘时进行了比较：
M1*M1*y=M2*y
双方获得相同的结果。因此，我认为我在其他地方遇到了问题，但我找不到它。
0 个答案:
没有答案
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