In the realm of numerical simulations and computational modeling. The choice between implicit and explicit solvers plays a crucial role in determining the accuracy, stability, and efficiency of the solutions obtained. These two distinct approaches offer unique advantages, and we employ this in various scientific and engineering applications.

In this blog, we will delve into the world of implicit and explicit solvers, exploring their characteristics, strengths, and weaknesses.

## Explicit Solvers

An explicit solver computes unknown variables at the next time step using only current time step information. It expresses the next time step values in terms of the current time step values. This makes it easier and less expensive than implicit methods. However, explicit methods can be less stable, necessitating smaller time steps for accuracy, particularly in certain problem types.

Example: The forward Euler method for time integration is an explicit solver. It updates the solution at time t+Δ*t* based on the value at time *t* and the derivative of the solution at time *t*.

## Implicit Solvers

An implicit solver, on the other hand, is a numerical method that involves solving equations or systems of equations that include both current and future values of the unknown variables. In other words, we define the solution at the next time step implicitly by an equation that involves the values at the current and future time steps. These methods can be more intensive because they require solving linear or non linear systems of equations. Still, they are generally more stable and allow for larger time steps, making them suitable for certain types of problems.

Example: Implicit methods like the backward Euler method for time integration define the solution at time t+Δ*t* in terms of the solution at t+Δ*t* and the derivative at t+Δ*t*. This results in an implicit equation that needs to be solved iteratively to obtain the updated solution.

The choice between implicit and explicit solvers depends on the specific problem, the desired level of accuracy, stability requirements, and computational resources available. Some problems may require a combination of both methods or the use of more specialized solvers to efficiently tackle different aspects of the simulation.

## Criteria for Stability

The stability conditions for both implicit and explicit solver schemes depend on the specific numerical methods used for time integration. Let’s discuss the stability conditions for two commonly used methods: the forward Euler method for explicit solvers and the backward Euler method for implicit solvers.

**Stability Condition for Explicit Solver**

The forward Euler method is a simple explicit time integration scheme. It updates the solution at the next time step based on the values at the current time step and the derivatives at the current time step. The stability condition for the forward Euler method is related to the step size (time step) used in the simulation.

The forward Euler method is conditionally stable, meaning it has a restriction on the time step size for stability. To maintain stability, the time step ($ \Delta t $) must be chosen such that it satisfies the following condition:

$$ \Delta t \leq \frac{C}{|\lambda|} $$

where (C) is a constant (usually less than 1) and ($|\lambda|$) is the maximum eigenvalue of the system matrix. If this condition is not met, the solution may become unstable, leading to inaccurate and divergent results.

**Stability Condition for Implicit Solver**

The backward Euler method is an implicit time integration scheme that updates the solution at the next time step based on the values at the next time step and the derivatives at the next time step. This method is unconditionally stable, meaning it does not impose any restrictions on the time step size for stability.

Unlike explicit methods, the backward Euler method considers future values, making it inherently stable regardless of the time step used. However, solving the implicit equation at each time step can be more expensive than explicit methods.

## Conclusions

In summary, the stability condition for explicit solvers like the forward Euler method requires a limitation on the time step size to ensure stability, while implicit solvers like the backward Euler method are unconditionally stable and do not impose any such restrictions. The choice between implicit and explicit solvers depends on the problem stability requirements, computational resources, and desired level of accuracy.

This article was crafted by a group of experts at eigenplus to ensure it adheres to our strict quality standards. The individuals who contributed to this article are: