An algebraic flux correction scheme facilitating the use of Newton-like solution strategies

Abstract

Building on recent advances in the analysis and design of algebraic flux correction (AFC) schemes, new differentiable limiter functions are constructed for efficient nonlinear solution strategies. The proposed scaling parameters are used to limit artificial diffusion operators incorporated into the residual of a high order target scheme to produce accurate and boundpreserving finite element approximations to hyperbolic problems. Due to this stabilization procedure, the occurring system becomes highly nonlinear and the efficient computation of corresponding solutions is a challenging task. The presented regularization approach makes the AFC residual twice continuously differentiable so that Newton's method converges quadratically for sufficiently good initial guesses. Furthermore, the performance of each nonlinear iteration is improved by expressing the Jacobian as the sum and product of matrices having the same sparsity pattern as the Galerkin system matrix. Eventually, the AFC methodology constructed for stationary problems is extended to transient test cases and validated numerically by applying it to several numerical benchmarks.

Algebraic flux correction finite element method with semi-implicit time stepping for solute transport in fractured porous media

“…In view of (31) we have that uij ≤ max(lij, lji), which implies that the last term in (33) is diffusive, in the sense that lij − uij ≥ 0 for j ∈ N + i . Now let us show that, for j ∈ N − i the anti-diffusive flux −uij(cj − ci) can be expressed as a sum of the diffusive ones, that is…”

“…Note that the scheme (38) relies on the linear low-order approximation approximation in the fracture domain. Alternatively one can replace the operator D f by the nonlinear one D f (c n+1 ) in which case the scheme would require to solve a nonlinear system at each time step [25], [20], [29], [21], [31]. However, since we are interested in using large time steps that would typically lead to the fracture CFL number much large than 1, it is not clear if such higher-order nonlinear implicit scheme would improve the accuracy.…”