Numerical Experiments

Routine Name: Numerical Experiments

Author: Kyle Hovey

Language: C++

Description/Purpose:

This code tests the accuracy of a given method against some examples found in the course textbook. In \(7.1\), we analyze \(u’ = -\sin(x); u_0 = 1; dt = 10^{-3}\). In \(7.2\), we change our driving function to \(u’ = \lambda (u - \cos(x)) - \sin(x)\) and require \(\lambda = -10\). And finally, for \(7.3\), we require \(\lambda = -2100\).

Input: A method for solving the examples

Output: The solutions given by the method

Usage/Example:

#include <iostream>
#include <cmath>
#include "../../euler/src/euler/euler.h"

int main() {
  /*
   * 7.1
   */
  const auto dt = 1E-6;
  const auto uo = 1;
  const auto simple = [](const double& t, const double& u) -> double {
    (void) u;
    return -std::sin(t);
  };
  const auto T = 2;

  auto soln = Euler::genExplicitEulerSolution<double>(simple, dt, uo);

  std::cout << "7.1:" << std::endl;
  std::cout << "U_2000 = approx(2): " << soln(T) << std::endl;
  std::cout << "Exact(2): " << std::cos(T) << std::endl;
  std::cout << "Error: " << std::abs(soln(T) - std::cos(T)) << std::endl;
  std::cout << std::endl;

  /*
   * 7.2
   */
  const auto modified = [](const double& t, const double& u) -> double {
    return -10 * (u - std::cos(t)) - std::sin(t);
  };

  soln = Euler::genExplicitEulerSolution<double>(modified, dt, uo);

  std::cout << "7.2:" << std::endl;
  std::cout << "U_2000 = approx(2): " << soln(T) << std::endl;
  std::cout << "Exact(2): " << std::cos(T) << std::endl;
  std::cout << "Error: " << std::abs(soln(T) - std::cos(T)) << std::endl;
  std::cout << std::endl;

  /*
   * 7.3
   */
  const auto largeLambda = [](const double& t, const double& u) -> double {
    return -2100 * (u - std::cos(t)) - std::sin(t);
  };

  soln = Euler::genExplicitEulerSolution<double>(largeLambda, dt, uo);

  std::cout << "7.3:" << std::endl;
  std::cout << "U_2000 = approx(2): " << soln(T) << std::endl;
  std::cout << "Exact(2): " << std::cos(T) << std::endl;
  std::cout << "Error: " << std::abs(soln(T) - std::cos(T)) << std::endl;

  return EXIT_SUCCESS;
}

Output:

Notice how the output for the first two examples has very high error. I’m still not completely sure why this happened (as there was very little error when testing the exact same code for the last assignment), but it is easy to see that the error blows up for \(7.3\) where \(\lambda = -21000\).

7.1:
U_2000 = approx(2): -0.818595
Exact(2): -0.416147
Error: 0.402448

7.2:
U_2000 = approx(2): -0.507077
Exact(2): -0.416147
Error: 0.0909297

7.3:
U_2000 = approx(2): 8.64194e+82
Exact(2): -0.416147
Error: 8.64194e+82

Implementation/Code:

All solver implementation for this assignment was covered in the last assignment.

Numerical Solutions (Math 5620)

Numerical Experiments