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.
翻译:Runge- Kutta (RK) 方案, 特别是 Gaus- Legendre 和其他一些完全隐含的 RK (FIRK) 方案, 因其稳定性和高度准确性, 对抛光部分偏差方程式的时间整合是可取的。 然而, 与对等隐含RK( 或DIRK) 方案相比, 为他们建立最佳的预设程序, 要比对等隐含RK( 或DIRK) 方案更具挑战性。 为了应对这一挑战, 我们首先引入数学上最佳的预设程序, 称为块状复合Schur分解( BCSD) 、 块真实正正正正正正正正方程式( BJ) 和 直立方程式( 以直立法( IMF) 等比等的存储和分解( IMF) 等值, 以直立法( IMRR) 等值( IM) 等值, 等值, 等值(WSDSD- Ral- Ral- IL) 等值, 和 等值, 等值, 等值, IM- RIS- L 等值, 等值, 等值, 等值, 等值, 等值。