Table of Contents

Heat Equation Implicit Euler

Routine Name: heat_implicit_euler

Author: Philip Nelson

Language: C++

Description

The heat equation is a linear partial differential equation

\[\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{ \partial x^2 }\]

that describes the distribution of heat (or variation in temperature) in a given region over time.

Input

heat_implicit_euler(T L, unsigned int Nx, T F, T a, T Tf)

• T L - Length of the domain ([0,L])
• unsigned int Nx - The total number of mesh points
• T F - The dimensionless number \(\alpha \cdot \frac{dt}{dx^2}\), which implicitly specifies the time step
• T a - Variable coefficient (constant)
• T Tf - The stop time for the simulation

Output

A vector containing the state of the system at \(Tf\).

Code

template <typename T>
auto heat_implicit_euler(T L, unsigned int Nx, T F, T a, T Tf)
{
  auto x = linspace<double>(0, L, Nx + 1);
  auto dx = x[1] - x[0];
  auto dt = F * dx * dx / a;
  auto Nt = std::round(Tf / dt);
  auto t = linspace<double>(0, Tf, Nt + 1);
  std::vector<T> u;
  std::vector<T> u_1;

  for (auto i = 0u; i <= Nx; ++i)
  {
    u.push_back(0);
    u_1.push_back(0);
  }

  u_1[Nx / 2] = 1000;

  for (auto n = 0u; n < Nt; ++n)
  {
    for (auto i = 1u; i < Nx; ++i)
    {
      u[i] = u_1[i] + F * (u_1[i - 1] - 2 * u_1[i] + u_1[i + 1]);
    }

    u[0] = 0;
    u[Nx] = 0;
    std::swap(u, u_1);
  }

  return u;
}

Example

int main()
{
  auto a = 0.7;
  auto F = 0.01;
  auto Nx = 10u;
  auto Tf = 1.0;
  auto L = 1.0;

  auto solution = heat_implicit_euler<double>(L, Nx, F, a, Tf);

  for(auto && x:solution)
    std::cout << x << '\n';

  return EXIT_SUCCESS;
}

Result

0
1.545
2.93876
4.04485
4.75501
4.99971
4.75501
4.04485
2.93876
1.545
0

Last Modification date: 5 May 2018