What distinguishes explicit methods from implicit methods in computational analysis?

The distinction between explicit and implicit methods in computational analysis primarily revolves around how they treat time integration and the dependencies of state variables at different time steps.

Explicit methods calculate the state of a system at the next time step solely based on information from the current state, which means they only require current time information. This approach involves directly substituting known values into equations to compute future values, making the implementation relatively straightforward. However, explicit methods can often face stability issues, especially with larger time steps, which can limit their applicability in certain scenarios.

On the other hand, implicit methods incorporate future state considerations. They require solving a system of equations that includes both the current and future states, as the state at the next time step is interdependent on the state of the system at that time. This approach often yields better stability properties, allowing for larger time steps without compromising accuracy, but it typically involves more complex computational effort due to the need for solving equations that can result from this interdependence.

Thus, the key distinction lies in the fact that explicit methods depend solely on current state information, while implicit methods require consideration of future states, thus impacting their implementation and the conditions under which they are used effectively.