The Numerical Solution of a Nonseparable Elliptic Partial Differential Equation by Preconditioned Conjugate Gradients

Area of Science:

• Numerical analysis
• Computational fluid dynamics
• Partial differential equations

Background:

• Solving linear algebraic equations from discretized partial differential equations is computationally intensive.
• Nonseparable elliptic partial differential equations require efficient solution methods.
• Buoyant convection in room fires involves complex fluid dynamics governed by elliptic equations.

Purpose of the Study:

• To present a combined iterative and direct method for solving nonseparable elliptic partial differential equations.
• To extend the preconditioned conjugate gradient technique to singular linear equations arising from Neumann boundary conditions.
• To apply these algorithms to model buoyant convection in room fires.

Main Methods:

• Combination of the conjugate gradient algorithm (iterative) with cyclic reduction (fast direct method).
• Development of a code to implement the combined algorithms.
• Application to a specific elliptic equation modeling buoyant convection.

Main Results:

• The developed code successfully implements the combined iterative and direct solution technique.
• New results demonstrate the efficacy of preconditioned conjugate gradients for singular linear equations with Neumann boundary conditions.
• Numerical results from the application to buoyant convection are discussed.

Conclusions:

• The combination of conjugate gradient and cyclic reduction offers an efficient approach for solving discretized nonseparable elliptic partial differential equations.
• The extended preconditioned conjugate gradient technique is effective for handling singular equations with Neumann boundary conditions.
• The implemented algorithms provide a valuable tool for simulating buoyant convection phenomena.