2 research outputs found
Linearly implicit GARK schemes
Systems driven by multiple physical processes are central to many areas of
science and engineering. Time discretization of multiphysics systems is
challenging, since different processes have different levels of stiffness and
characteristic time scales. The multimethod approach discretizes each physical
process with an appropriate numerical method; the methods are coupled
appropriately such that the overall solution has the desired accuracy and
stability properties. The authors developed the general-structure additive
Runge-Kutta (GARK) framework, which constructs multimethods based on
Runge-Kutta schemes.
This paper constructs the new GARK-ROS/GARK-ROW families of multimethods
based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary
differential equation models, we develop a general order condition theory for
linearly implicit methods with any number of partitions, using exact or
approximate Jacobians. We generalize the order condition theory to two-way
partitioned index-1 differential-algebraic equations. Applications of the
framework include decoupled linearly implicit, linearly implicit/explicit, and
linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of
order up to four are constructed
Linearly Implicit Multistep Methods for Time Integration
Time integration methods for solving initial value problems are an important
component of many scientific and engineering simulations. Implicit time
integrators are desirable for their stability properties, significantly
relaxing restrictions on timestep size. However, implicit methods require
solutions to one or more systems of nonlinear equations at each timestep, which
for large simulations can be prohibitively expensive. This paper introduces a
new family of linearly implicit multistep methods (LIMM), which only requires
the solution of one linear system per timestep. Order conditions and stability
theory for these methods are presented, as well as design and implementation
considerations. Practical methods of order up to five are developed that have
similar error coefficients, but improved stability regions, when compared to
the widely used BDF methods. Numerical testing of a self-starting variable
stepsize and variable order implementation of the new LIMM methods shows
measurable performance improvement over a similar BDF implementation.Comment: 36 pages, 5 figures, submitted to SISC in May 202