Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs
نویسندگان
چکیده
Recently, the authors introduced a preconditioner for the linear systems that arise from fully implicit Runge-Kutta time stepping schemes applied to parabolic PDEs [9]. The preconditioner was a block Jacobi preconditioner, where each of the blocks were based on standard preconditioners for low-order time discretizations like implicit Euler or Crank-Nicolson. It was proven that the preconditioner is optimal with respect to the timestep and the discretization parameter in space. In this paper we will improve the convergence by considering other preconditioners like the upper and the lower block Gauss-Seidel preconditioners, both in a left and right preconditioning setting. Finally, we improve the condition number by using a generalized Gauss-Seidel preconditioner.
منابع مشابه
Order-Optimal Preconditioners for Implicit Runge-Kutta Schemes Applied to Parabolic PDEs
In this paper we show that standard preconditioners for parabolic PDEs discretized by implicit Euler or Crank–Nicolson schemes can be reused for higher–order fully implicit Runge–Kutta time discretization schemes. We prove that the suggested block diagonal preconditioners are order–optimal for A–stable and irreducible Runge–Kutta schemes with invertible coefficient matrices. The theoretical inv...
متن کاملSpectral Deferred Corrections for Parabolic Partial Differential Equations
tions (PDEs). This class of schemes is based on three principal observations. First, the spatial discretization of parabolic PDEs results in stiff systems of ordinary differential equations (ODEs) in time, and therefore, requires an implicit method for its solution. Spectral Deferred Correction (SDC) methods use repeated iterations of a low-order method (e.g. implicit Euler method) to generate ...
متن کاملOn Runge-Kutta Methods for Parabolic Problems with Time-Dependent Coefficients
Galerkin fully discrete approximations for parabolic equations with time-dependent coefficients are analyzed. The schemes are based on implicit Runge-Kutta methods, and are coupled with preconditioned iterative methods to approximately solve the resulting systems of linear equations. It is shown that for certain classes of Runge-Kutta methods, the fully discrete equations exhibit parallel featu...
متن کاملLinearly implicit methods for nonlinear parabolic equations
We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes...
متن کاملMultigrid Methods for Implicit Runge-Kutta and Boundary Value Method Discretizations of Parabolic PDEs
Sophisticated high order time discretization methods, such as implicit Runge–Kutta and boundary value methods, are often disregarded when solving time dependent partial differential equations, despite several appealing properties. This is mainly because it is considered hard to develop efficient methods for the more complex linear systems involved. We show here that for implicit Runge–Kutta and...
متن کامل