TY - JOUR
T1 - Reliability Investigation of BiCGStab and IDR Solvers for the Advection-Diffusion-Reaction Equation
AU - Schoutrop , Chris
AU - Boonkkamp , Jan ten Thije
AU - Dijk , Jan van
JO - Communications in Computational Physics
VL - 1
SP - 156
EP - 188
PY - 2022
DA - 2022/07
SN - 32
DO - http://doi.org/10.4208/cicp.OA-2021-0182
UR - https://global-sci.org/intro/article_detail/cicp/20791.html
KW - BiCGStab, IDR, shadow residual, advection-diffusion-reaction equation.
AB - <p style="text-align: justify;">The reliability of BiCGStab and IDR solvers for the exponential scheme discretization of the advection-diffusion-reaction equation is investigated. The resulting discretization matrices have real eigenvalues. We consider BiCGStab, IDR$(S),$ BiCGStab$(L)$ and various modifications of BiCGStab, where $S$ denotes the dimension of
the shadow space and $L$ the degree of the polynomial used in the polynomial part. Several implementations of BiCGStab exist which are equivalent in exact arithmetic, however, not in finite precision arithmetic. The modifications of BiCGStab we consider are;
choosing a random shadow vector, a reliable updating scheme, and storing the best intermediate solution. It is shown that the Local Minimal Residual algorithm, a method
similar to the &quot;minimize residual&quot; step of BiCGStab, can be interpreted in terms of
a time-dependent advection-diffusion-reaction equation with homogeneous Dirichlet
boundary conditions for the residual, which plays a key role in the convergence analysis. Due to the real eigenvalues, the benefit of BiCGStab$(L)$ compared to BiCGStab
is shown to be modest in numerical experiments. Non-sparse (e.g. uniform random)
shadow residual turns out to be essential for the reliability of BiCGStab. The reliable
updating scheme ensures the required tolerance is truly achieved. Keeping the best intermediate solution has no significant effect. Recommendation is to modify BiCGStab
with a random shadow residual and the reliable updating scheme, especially in the
regime of large Péclet and small Damköhler numbers. An alternative option is IDR($S$),
which outperforms BiCGStab for problems with strong advection in terms of the number of matrix-vector products. The MATLAB code used in the numerical experiments is available on GitLab: https://gitlab.com/ChrisSchoutrop/krylov-adr, a
C++ implementation of IDR$(S)$ is available in the Eigen linear algebra library: http:
//eigen.tuxfamily.org.</p>