Structure preserving iterative methods in CFD

Viktor Linders
Lund University


Abstract:

The discrete preservation of linear and nonlinear physical invariants, such as conservation of mass, energy and entropy, form the backbone of robust high order methods in CFD. For stiff problems, e.g. wall-bounded flows, these methods are necessarily implicit, resulting in large systems of non-linear equations. Solutions to these systems are typically approximated using iterative methods. However, since iterative methods in practice are truncated well before asymptotic convergence is reached, the approximate solution may violate physical invariants.

In this talk, we explore the possibility of preserving certain invariants with iterative methods for linear and nonlinear problems. We discuss the conservation of mass, noting that common methods such as Jacobi and Gauss-Seidel are not conservative. Local conservation and the Lax-Wendroff theorem are investigated for pseudo-time stepping schemes and it is found that they may converge to incorrect solutions. For a nonlinear problem it is demonstrated that Newton's method violates entropy conservation. A variety of approaches towards a remedy are discussed.