
Linearly Implicit Multistep Methods for Time Integration
Time integration methods for solving initial value problems are an impor...
read it

Linearized Implicit Methods Based on a SingleLayer Neural Network: Application to KellerSegel Models
This paper is concerned with numerical approximation of some twodimensi...
read it

A New ImplicitExplicit Local Method to Capture Stiff Behavior with COVID19 Outbreak Application
In this paper, a new implicitexplicit local method with an arbitrary or...
read it

Adaptive KrylovType Time Integration Methods
The RosenbrockKrylov family of time integration schemes is an extension...
read it

EPIRKW and EPIRKK time discretization methods
Exponential integrators are special time discretization methods where th...
read it

A New Block Preconditioner for Implicit RungeKutta Methods for Parabolic PDE
A new preconditioner based on a block LDU factorization with algebraic m...
read it

Numerical Stability of Tangents and Adjoints of Implicit Functions
We investigate errors in tangents and adjoints of implicit functions res...
read it
Biorthogonal RosenbrockKrylov time discretization methods
Many scientific applications require the solution of large initialvalue problems, such as those produced by the method of lines after semidiscretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed RosenbrockKrylov (ROK) time integration methods extend the classical linearlyimplicit Rosenbrock(W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short twoterm recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.
READ FULL TEXT
Comments
There are no comments yet.