Explicit schemes

Definition

Explicit schemes are numerical methods used to solve partial differential equations by updating the solution at each time step based solely on known information from the previous time step. They are straightforward in implementation, making them popular for various applications in computational fluid dynamics. However, explicit schemes require careful consideration of stability conditions, often governed by a criterion like the Courant-Friedrichs-Lewy (CFL) condition.

5 Must Know Facts For Your Next Test

1. Explicit schemes are conditionally stable, meaning they can only be used with specific time step sizes related to the spatial grid size.
2. The CFL condition is a critical criterion determining the maximum allowable time step in explicit schemes to ensure stability.
3. They are often easier to implement than implicit schemes since explicit schemes do not require solving a system of equations at each time step.
4. Explicit schemes can be less accurate than implicit ones when dealing with stiff problems due to their restrictions on time step size.
5. Common examples of explicit schemes include the Forward Euler method and various finite difference methods.

Review Questions

• How does the CFL condition influence the choice of time step when using explicit schemes?
  • The CFL condition serves as a guideline to ensure stability in explicit schemes. It relates the time step size to the spatial discretization, requiring that the wave propagation speed does not exceed a certain threshold. If the chosen time step violates this condition, it can lead to unbounded growth of errors in the solution, causing numerical instability. Therefore, adhering to the CFL condition is crucial for effective implementation of explicit schemes.
• Compare explicit and implicit schemes in terms of stability and computational complexity.
  • Explicit schemes are generally simpler to implement and computationally less intensive since they do not require solving a system of equations at each time step. However, they face limitations in stability and can only use smaller time steps dictated by the CFL condition. In contrast, implicit schemes allow for larger time steps and can handle stiff problems better, but they involve more complex calculations and often require iterative solvers, making them computationally more demanding.
• Evaluate the suitability of explicit schemes for solving real-world fluid dynamics problems with regard to accuracy and efficiency.
  • Explicit schemes can be very effective for certain fluid dynamics problems where accuracy is critical and the problem does not exhibit stiff behavior. Their ease of implementation and ability to provide quick results makes them suitable for simulations involving complex geometries or transient flows. However, their reliance on smaller time steps limits their efficiency when dealing with high-speed flows or problems requiring long simulation times. Therefore, while explicit schemes have their place in fluid dynamics modeling, understanding their limitations is key to selecting the right approach for specific problems.

Related terms

Implicit schemes: Numerical methods that update the solution by solving equations involving future time steps, allowing for larger time steps but often requiring more complex calculations.

Stability analysis: The study of whether small changes in initial conditions lead to bounded or unbounded solutions over time, crucial for assessing the performance of numerical schemes.

Finite volume method: A numerical technique that conserves fluxes through a control volume, ideal for solving conservation laws and widely used in fluid dynamics.