# Fast Solution of Fully Implicit Runge-Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting

@article{Southworth2022FastSO, title={Fast Solution of Fully Implicit Runge-Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting}, author={Ben S. Southworth and Oliver A. Krzysik and Will Pazner and Hans De Sterck}, journal={SIAM J. Sci. Comput.}, year={2022}, volume={44}, pages={416-} }

Fully implicit Runge-Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but are rarely used in practice with numerical PDEs due to the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic preconditioning framework for solving the systems of equations that arise from IRK methods applied to linear numerical PDEs (without algebraic constraints). This framework also naturally applies to…

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## 4 Citations

Fast Solution of Fully Implicit Runge-Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

- Computer ScienceSIAM J. Sci. Comput.
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A theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDE's with algebraic constraints is introduced.

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This work considers strongly A-stable implicit Runge-Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for the arising large algebraic systems and introduces an efficient preconditioner, that allows for fully stage-parallel solution.

Weighted-norm preconditioners for a multi-layer tide model

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A robust method is derived that requires solving a reformulated system that only involves coupling between adjacent layers, and is based on a careful analysis of the matrix that couples the layers.

Arbitrary Order Energy and Enstrophy Conserving Finite Element Methods for 2D Incompressible Fluid Dynamics and Drift-Reduced Magnetohydrodynamics

- PhysicsArXiv
- 2022

Maintaining conservation laws in the fully discrete setting is critical for accurate long-time behavior of numerical simulations and requires accounting for discrete conservation properties in both…

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