PhD defense Fatemeh Mohammadi
"Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints"
Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations
(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).
In this thesis we study the initialization of multistep methods and parametrize some well-known classes
of multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)
multistep methods and parametric formulation of $\beta-$blocked multistep methods.
Depending on the number of steps, a multistep method requires adequate number of initial values to
start the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.
The proposed starters estimate the error by embedded methods.
The second part concerns the variable step-size $\beta-$blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize $\beta-$blocked multistep methods for
the solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments.
For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulation
allows time adaptivity by construction.