Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but are rarely used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for the fast, parallel solution of the nonlinear equations that arise from IRK methods  applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. This framework also naturally applies to discontinuous Galerkin discretizations in time. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2x2 systems. Under quite general assumptions on the spatial discretization , it is proven that the preconditioned  operator has a condition number of ~O(1),  with only weak dependence on the number of stages or integration accuracy.  The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss integration , while in all cases 4th-order Gauss integration requires roughly half the number of preconditioner applications as required by standard SDIRK methods.
Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but are rarely used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces 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 PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2x2 systems. Under quite general assumptions , it is proven that the preconditioned 2x2 operator has a condition number of ~O(1), independent of the spatial discretization, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK , while in all cases 4th-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard SDIRK methods.
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the fast, parallel solution of the
solving the
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(and discontinuous Galerkin discretizations in time)
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This framework also naturally applies to discontinuous Galerkin discretizations in time. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes.
D
on the spatial discretization
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2x2
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independent of the spatial discretization, and
A
Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes.
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integration
IRK
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integration
IRK