[Paper Review] Variational structure of the optimal artificial diffusion method for the advection-diffusion equation

K. B. Nakshatrala, Albert J. Valocchi|arXiv (Cornell University)|May 29, 2009

Advanced Numerical Methods in Computational Mathematics8 references3 citations

TL;DR

This paper establishes a variational foundation for the optimal artificial diffusion method in advection-diffusion problems by deriving a stable weak formulation from a variational principle. The method produces exact nodal solutions in one-dimensional problems with constant coefficients and forcing, proving that the optimal artificial diffusion method is equivalent to a variational formulation based on a weighted inner product with an exponential weight function.

ABSTRACT

In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions when applied to one-dimensional problems with constant coefficients and forcing function. We first present a variational principle for a multi-dimensional advective-diffusive system, and then derive a new stable weak formulation. When applied to one-dimensional problems with constant coefficients and forcing function, this resulting weak formulation will be equivalent to the optimal artificial diffusion method. We present representative numerical results to corroborate our theoretical findings.

Motivation & Objective

• To provide a variational foundation for the optimal artificial diffusion method, which previously lacked a variational formulation.
• To address the instability and oscillations in classical Galerkin methods for advection-dominated problems.
• To derive a stable weak formulation for the advection-diffusion equation using a variational principle.
• To demonstrate that the derived formulation yields the same difference equation as the optimal artificial diffusion method in one-dimensional cases.
• To highlight the theoretical significance of variational principles for non-self-adjoint operators like the advection-diffusion operator.

Proposed method

• Formulates the advection-diffusion equation as a non-self-adjoint operator and derives a variational principle using a weighted inner product with a weight function α(x) = exp(−vx/k).
• Applies Vainberg’s theorem to connect the weighted residual statement to a scalar functional, enabling a variational formulation.
• Derives a stable weak formulation by minimizing a functional involving the weighted inner product, ensuring stability for advection-dominated problems.
• Uses linear finite elements with shape functions defined over a three-node stencil to derive the discrete difference equation.
• Introduces a weight function α(x) that depends on the Péclet number, ensuring stability and nodal exactness in 1D.
• Compares the resulting discrete system with the classical optimal artificial diffusion method to confirm equivalence in 1D.

Experimental results

Research questions

• RQ1 Can the optimal artificial diffusion method be derived from a variational principle?
• RQ2 Is there a stable weak formulation for the advection-diffusion equation that produces exact nodal solutions in 1D with constant coefficients?
• RQ3 Can a scalar functional be constructed for the advection-diffusion operator using Vainberg’s theorem?
• RQ4 Does the derived variational formulation yield the same discrete system as the optimal artificial diffusion method in 1D?
• RQ5 What is the theoretical significance of using a non-uniform weight function in the variational formulation for non-self-adjoint problems?

Key findings

• The stable weak formulation derived from the variational principle produces the same difference equation as the optimal artificial diffusion method for one-dimensional advection-diffusion problems with constant coefficients and forcing.
• The coefficients in the resulting discrete system match exactly with those of the optimal artificial diffusion method: γ−1 = −v/(2h)(1 + coth(Ph/e)), γ0 = v/h coth(Ph/e), γ+1 = v/(2h)(1 − coth(Ph/e)).
• Numerical results confirm that the new formulation achieves nodally exact solutions for all Péclet numbers in 1D, validating the theoretical equivalence.
• The optimal artificial diffusion method is shown to have a firm variational basis through the derived functional and weight function α(x) = exp(−vx/k).
• The method is theoretically robust and provides a foundation for developing new stabilized formulations, though it may be computationally expensive for large-scale 2D and 3D problems.
• The study highlights the relevance of Tonti’s theoretical work on variational principles for non-self-adjoint systems in the context of stabilized finite element methods.