Perturbed Strong Stability Preserving Time-Stepping Methods For Hyperbolic PDEs
A plethora of physical phenomena are modelled by hyperbolic partial differential equations, for which the exact solution is usually not known. Numerical methods are employed to approximate the solution to hyperbolic problems; however, in many cases it is difficult to satisfy certain physical prop...
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| Language: | en |
| Published: |
2017
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| Online Access: | http://hdl.handle.net/10754/625526 http://repository.kaust.edu.sa/kaust/handle/10754/625526 |
| Summary: | A plethora of physical phenomena are modelled by hyperbolic partial differential
equations, for which the exact solution is usually not known. Numerical methods
are employed to approximate the solution to hyperbolic problems; however, in many
cases it is difficult to satisfy certain physical properties while maintaining high order
of accuracy. In this thesis, we develop high-order time-stepping methods that
are capable of maintaining stability constraints of the solution, when coupled with
suitable spatial discretizations. Such methods are called strong stability preserving
(SSP) time integrators, and we mainly focus on perturbed methods that use both
upwind- and downwind-biased spatial discretizations.
Firstly, we introduce a new family of third-order implicit Runge–Kuttas methods
with arbitrarily large SSP coefficient. We investigate the stability and accuracy of
these methods and we show that they perform well on hyperbolic problems with large
CFL numbers. Moreover, we extend the analysis of SSP linear multistep methods to
semi-discretized problems for which different terms on the right-hand side of the
initial value problem satisfy different forward Euler (or circle) conditions. Optimal
perturbed and additive monotonicity-preserving linear multistep methods are studied
in the context of such problems. Optimal perturbed methods attain augmented
monotonicity-preserving step sizes when the different forward Euler conditions are
taken into account. On the other hand, we show that optimal SSP additive methods achieve a monotonicity-preserving step-size restriction no better than that of the corresponding
non-additive SSP linear multistep methods. Furthermore, we develop the
first SSP linear multistep methods of order two and three with variable step size, and
study their optimality. We describe an optimal step-size strategy and demonstrate
the effectiveness of these methods on various one- and multi-dimensional problems.
Finally, we establish necessary conditions to preserve the total variation of the solution
obtained when perturbed methods are applied to boundary value problems.
We implement a stable treatment of nonreflecting boundary conditions for hyperbolic
problems that allows high order of accuracy and controls spurious wave reflections.
Numerical examples with high-order perturbed Runge–Kutta methods reveal that this
technique provides a significant improvement in accuracy compared with zero-order
extrapolation. |
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