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TheAlgorithms/C++: numerical_methods/ode_semi_implicit_euler.cpp File Reference

Solve a multivariable first order ordinary differential equation (ODEs) using semi implicit Euler method More...

#include <cmath>
#include <ctime>
#include <fstream>
#include <iostream>
#include <valarray>

Go to the source code of this file.

Solve a multivariable first order ordinary differential equation (ODEs) using semi implicit Euler method

The ODE being solved is:

\begin{eqnarray*} \dot{u} &=& v\\ \dot{v} &=& -\omega^2 u\\ \omega &=& 1\\ [x_0, u_0, v_0] &=& [0,1,0]\qquad\ldots\text{(initial values)} \end{eqnarray*}

The exact solution for the above problem is:

\begin{eqnarray*} u(x) &=& \cos(x)\\ v(x) &=& -\sin(x)\\ \end{eqnarray*}

The computation results are stored to a text file semi_implicit_euler.csv and the exact soltuion results in exact.csv for comparison.

To implement Van der Pol oscillator, change the problem function to:

const double mu = 2.0;

dy[0] = y[1];

dy[1] = mu * (1.f - y[0] * y[0]) * y[1] - y[0];

See also

ode_midpoint_euler.cpp, ode_forward_euler.cpp

Definition in file ode_semi_implicit_euler.cpp.

Exact solution of the problem. Used for solution comparison.

Parameters

[in] x independent variable [in,out] y dependent variable

Definition at line 66 of file ode_semi_implicit_euler.cpp.

66 {

67 y[0][0] = std::cos(x);

68 y[0][1] = -std::sin(x);

69}

Main Function

Definition at line 189 of file ode_semi_implicit_euler.cpp.

189 {

190 double X0 = 0.f;

191 double X_MAX = 10.F;

192 std::valarray<double> Y0 = {1.f, 0.f};

193 double step_size;

194

195 if (argc == 1) {

196 std::cout << "\nEnter the step size: ";

197 std::cin >> step_size;

198 } else {

199

200 step_size = std::atof(argv[1]);

201 }

202

203

205 std::cout << "\tTime = " << total_time << " ms\n";

206

207

209

210 return 0;

211}

double semi_implicit_euler(double dx, double x0, double x_max, std::valarray< double > *y, bool save_to_file=false)

Compute approximation using the semi-implicit-Euler method in the given limits.

void save_exact_solution(const double &X0, const double &X_MAX, const double &step_size, const std::valarray< double > &Y0)

Problem statement for a system with first-order differential equations. Updates the system differential variables.

Note

This function can be updated to and ode of any order.

Parameters

[in] x independent variable(s) [in,out] y dependent variable(s) [in,out] dy first-derivative of dependent variable(s)

Definition at line 53 of file ode_semi_implicit_euler.cpp.

54 {

55 const double omega = 1.F;

56 dy[0][0] = y[0][1];

57 dy[0][1] = -omega * omega * y[0][0];

58}

Function to compute and save exact solution for comparison

Parameters

[in] X0 initial value of independent variable [in] X_MAX final value of independent variable [in] step_size independent variable step size [in] Y0 initial values of dependent variables

Definition at line 153 of file ode_semi_implicit_euler.cpp.

155 {

156 double x = X0;

157 std::valarray<double> y = Y0;

158

159 std::ofstream fp("exact.csv", std::ostream::out);

160 if (!fp.is_open()) {

161 std::perror("Error! ");

162 return;

163 }

164 std::cout << "Finding exact solution\n";

165

166 std::clock_t t1 = std::clock();

167 do {

168 fp << x << ",";

169 for (int i = 0; i < y.size() - 1; i++) {

170 fp << y[i] << ",";

171 }

172 fp << y[y.size() - 1] << "\n";

173

175

176 x += step_size;

177 } while (x <= X_MAX);

178

179 std::clock_t t2 = std::clock();

180 double total_time = static_cast<double>(t2 - t1) / CLOCKS_PER_SEC;

181 std::cout << "\tTime = " << total_time << " ms\n";

182

183 fp.close();

184}

void exact_solution(const double &x, std::valarray< double > *y)

Exact solution of the problem. Used for solution comparison.

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