Overview of ODE-IVP solvers in MATLAB

We next provide an overview of ODE-IVP solvers. The contents of this section provide a sufficient background to solve problems of the form 

we describe the basic time-marching approach of ODE-IVP solvers and contrast explicit and implicit, single-step and multistep solvers. Then, we demonstrate the use of the explicit single-step solver ode45 and the imphcit multistep solver odel 5s. 

ODE solvers update x(t) in discrete time steps of size At to compute x(tk) at times to h 2 For a constant time step, tk = to + k(At) but often At varies throughout the course of the simulation. At is smaller for greater accuracy when x(t) changes rapidly and is larger for increased simulation speed when x(t) changes slowly. Over each time step, the exact update of the state vector is 

115) uses information only about the state values at the beginning, xl l, and end, xl + l, of the current time step, it is said to define a single-step integration method. For example, in the Crank-Nicholson method 

Because fix 0) generally is nonlinear, (4.115) often cannot be rearranged to provide a direct expression for x. Then, (4.115) is said to generate an implicit integration method that requires a nonlinear algebraic system to be solved at each time step. 

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