Cauchy Problem Solver

This project defines a Cauchy problem and provides various methods to calculate, visualize, and save numerical solutions. The CauchyProblem class is designed specifically for defining a Cauchy problem, with parameters that describe the mathematical problem remaining unmodifiable. However, parameters related to the numerical solution, such as the number of steps and the theta parameter, can be adjusted.

Features

The CauchyProblem class implements four numerical schemes to solve the problem:

• Forward Euler: EFSolution()
• Backward Euler: EBSolution()
• Crank-Nicolson: CNSolution()
• General Theta Scheme: ThetaSolution(theta)

Each method returns the solution as a NumSolution, which is defined as a std::tuple<std::vector<double>, std::vector<double>>.

• The first vector contains the time points at which the solution is evaluated.
• The second vector contains the corresponding values of the solution at those time points.

Solution Storage

Each solution calculated using one of the four methods is saved in an internal array (Solutions) to avoid redundant recalculations with the same method. In particular, for ThetaSolution, the solution is stored for the most recent value of theta. If theta remains unchanged, the solution is not recalculated.

Visualization and Output

The CauchyProblem class provides three methods to manage the calculated solution:

• Print: print() outputs the solution to the console.
• Plot: plot() visualizes the solution using Gnuplot.
• Save: save() saves the solution to a file.

By default, these methods use the most recently calculated solution. You can select a different solution using SetTheta(theta). If the requested solution has not yet been computed, it will be calculated automatically when needed.

Constructors

The class provides two constructors:

1. CauchyProblem(ODE f, double y0, double t0, double t1) – Fully defines the Cauchy problem.
2. CauchyProblem(ODE f) – Requires only the ODE to be solved, and defaults to y0 = 0, t0 = 0, T = 1.

Solver Class

The Solver class calculates the solution using the finite difference method with a general theta parameter. For root-finding within the solver, the secant method implemented in basicZeroFun.hpp is used. To compute the derivative, the endpoints of the interval centered at the previous result u[i-1] are used, with a width equal to 2*(2*odefun(t[i],u[i-1])+0.01).

Requirements

• You must integrate the project with the repository pacs-examples: Set the path to pacs-examples in the PATH_PACSEXAMPLE variable in the Makefile. This is necessary because the project uses the basicZeroFun.hpp file and its associated libraries.
• The Boost library is required for using Gnuplot to visualize the results.

Compilation

To compile the project, follow these steps:

1. Clone the pacs-examples repository:

   git clone https://github.com/pacs-course/pacs-examples.git