Implicit sampling methods and explicit sampling methods are two different approaches used in numerical simulations and analyses, particularly in solving differential equations. The primary difference lies in how they treat the relationship between variables in the equations, especially when determining the state of a system at the next time step.
In explicit sampling methods, the future state of a system is calculated directly from the current state. This means that you can compute the next value using a formula that only relies on known values from the present. A common example of an explicit method is the Euler method for ordinary differential equations. For instance, if you have a simple equation that models the position of an object under constant acceleration, you can straightforwardly calculate the future position using the current position and velocity. The ease of implementation is one of the significant advantages of explicit methods, along with their ability to be computationally efficient for simple problems.
On the other hand, implicit sampling methods require solving an equation that relates both the current and future states. This means you often need to rearrange or solve a system of equations to find the next values. For example, in implicit methods like the backward Euler method, the future state is linked to the current state in a way that requires additional computation to determine the next step. This can involve solving nonlinear equations, making implicit methods generally more complex but often more stable, especially for stiff equations. Even though they can be more computationally demanding, implicit methods tend to provide better stability for systems that exhibit rapid changes, which is essential in many engineering applications.