Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal and Low-Memory Near-Optimal Preconditioning of Fully Implicit
  Runge-Kutta Schemes for Parabolic PDEs   Runge-Kutta (RK) schemes, especially Gauss-Legendre and some other fully
implicit RK (FIRK) schemes, are desirable for the time integration of parabolic
partial differential equations due to their A-stability and high-order
accuracy. However, it is significantly more challenging to construct optimal
preconditioners for them compared to diagonally implicit RK (or DIRK) schemes.
To address this challenge, we first introduce mathematically optimal
preconditioners called block complex Schur decomposition (BCSD), block real
Schur decomposition (BRSD), and block Jordan form (BJF), motivated by
block-circulant preconditioners and Jordan form solution techniques for IRK. We
then derive an efficient, near-optimal singly-diagonal approximate BRSD
(SABRSD) by approximating the quasi-triangular matrix in real Schur
decomposition using an optimized upper-triangular matrix with a single diagonal
value. A desirable feature of SABRSD is that it has comparable memory
requirements and factorization (or setup) cost as singly DIRK (SDIRK). We
approximate the diagonal blocks in these preconditioning techniques using an
incomplete factorization with (near) linear complexity, such as multilevel ILU,
ILU(0), or a multigrid method with an ILU-based smoother. We apply the block
preconditioners in right-preconditioned GMRES to solve the advection-diffusion
equation in 3D using finite element and finite difference methods. We show that
BCSD, BRSD, and BJF significantly outperform other preconditioners in terms of
GMRES iterations, and SABRSD is competitive with them and the prior state of
the art in terms of computational cost while requiring the least amount of
memory.

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The categories of this paper are: econ
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph