Conservative iterative methods for implicit discretizations of conservation laws

by   Philipp Birken, et al.

Conservation properties of iterative methods applied to implicit finite volume discretizations of nonlinear conservation laws are analyzed. It is shown that any consistent multistep or Runge-Kutta method is globally conservative. Further, it is shown that Newton's method, Krylov subspace methods and pseudo-time iterations are globally conservative while the Jacobi and Gauss-Seidel methods are not in general. If pseudo-time iterations using an explicit Runge-Kutta method are applied to a locally conservative discretization, then the resulting scheme is also locally conservative. However, the corresponding numerical flux can be inconsistent with the conservation law. We prove an extension of the Lax-Wendroff theorem, which reveals that numerical solutions based on these methods converge to weak solutions of a modified conservation law where the flux function is multiplied by a particular constant. This constant depends on the choice of Runge-Kutta method but is independent of both the conservation law and the discretization. Consistency is maintained by ensuring that this constant equals unity and a strategy for achieving this is presented. Experiments show that this strategy improves the convergence rate of the pseudo-time iterations.


page 1

page 2

page 3

page 4


Locally conservative and flux consistent iterative methods

Conservation and consistency are fundamental properties of discretizatio...

Entropy stable reduced order modeling of nonlinear conservation laws

Reduced order models of nonlinear conservation laws in fluid dynamics do...

Resolving Entropy Growth from Iterative Methods

We consider entropy conservative and dissipative discretizations of nonl...

A note on the conservation properties of the generalized-α method

We show that the second-order accurate generalized-α method on a uniform...

Numerical conservative solutions of the Hunter–Saxton equation

We derive a convergent (up to a subsequence) numerical method for conser...

On the convergence rate of the Kačanov scheme for shear-thinning fluids

We explore the convergence rate of the Kačanov iteration scheme for diff...

Please sign up or login with your details

Forgot password? Click here to reset