Solver

• Calculate the next states values in a single step using the system differential equations and the current state values

• Unstable when used to solve a stiff system unless its time step is set to a prohibitively small value

stiff —> ODE contains both rapidly changing and slowly changing variables

Example is Forward Euler:

explicit because the new state, y(n+1), is computed directly from the current state, y(n), without solving a nonlinear equation system.

(In the Forward Euler method, the solution of the differential equation is approximated using a Taylor series expansion around the current time step. The approximation is then truncated at the first order, and the resulting formula is used to compute the next state of the system.)

ODE23 and ODE45 are two different solvers used for numerical integration of ordinary differential equations (ODEs) in MATLAB.

ODE23 is a low-order solver that uses a second and third-order Runge-Kutta method to approximate the solution of an ODE. This solver is generally faster than ODE45 for problems that require less accuracy, but it may not be as accurate as ODE45.

On the other hand, ODE45 is a higher-order solver that uses a fourth and fifth-order Runge-Kutta method to approximate the solution of an ODE. This solver is more accurate than ODE23 and is suitable for problems that require higher accuracy. However, it is also slower than ODE23, especially for large systems of ODEs.

Therefore, the choice between ODE23 and ODE45 depends on the specific problem at hand. If speed is more important than accuracy, ODE23 may be the better choice. If accuracy is more important, or if the problem requires a higher order solver, ODE45 may be the better choice.

• Require multiple iteration steps to calculate the next state values

• Iterative Estimation of the next state to calculate the next state until the estimate is equal to the calculated result

• More computations less efficient than a non‐stiff solver

• Stability region is larger than that of a non‐stiff solver

• Larger time steps can be taken without encountering instability problems

Example is backward Euler:

• Stability region is larger than that of a explicit solver

• Larger time steps can be taken without encountering instability problems

(In summary, the larger stability region of implicit solvers is due to their ability to use future information to maintain stability, which makes them more accurate and efficient than explicit solvers, especially for stiff systems.)

A variable step solver is a type of numerical solver used to approximate the solution of ordinary differential equations (ODEs). In contrast to fixed step solvers, which use a fixed step size to advance the solution in time, variable step solvers adjust the step size based on the local behavior of the solution.

The basic idea of a variable step solver is to use a smaller step size in regions where the solution is changing rapidly, and a larger step size in regions where the solution is changing slowly. This approach can help to improve the accuracy of the solution while also reducing the computational cost.

One common example of a variable step solver is the ODE45, which uses a fourth or fifth-order Runge-Kutta method to approximate the solution at a specific time and adjusts the step size based on the local error in the solution. If the error estimate is larger than a specified tolerance, the solver reduces the step size. If the error estimate is smaller than the tolerance, the solver increases the step size and continues with the calculation.

Overall, variable step solvers are a powerful tool for solving ODEs, particularly in situations where the solution varies rapidly over some parts of the domain and more slowly over others. By adjusting the step size dynamically, these solvers can provide accurate solutions with a minimum of computational cost.

The Runge-Kutta method is a numerical technique for solving ordinary differential equations (ODEs) numerically. The method was developed by Carl Runge and Martin Kutta in the late 19th century.

The basic idea of the Runge-Kutta method is to use a weighted average of several estimates of the derivative of the solution to approximate the next value of the solution. In other words, the method uses a weighted combination of different approximations to the derivative of the solution at different points in the interval being considered.

There are many variations of the Runge-Kutta method, with different orders of accuracy and different numbers of stages. The most commonly used variant is the fourth-order Runge-Kutta method, also known as RK4. This method involves computing four different estimates of the derivative of the solution at different points in the interval and using a weighted average of these estimates to approximate the next value of the solution.

The Runge-Kutta method is widely used in numerical analysis and scientific computing to solve ODEs that cannot be solved analytically. It is a powerful and versatile method that can be used to solve a wide range of problems in physics, engineering, and other scientific disciplines.