Reliability Investigation of BiCGStab and IDR Solvers for the Advection-Diffusion-Reaction Equation

Abstract

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 ''minimize residual" 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 Damk öhler 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.