A plethora of physical phenomena are modelled by hyperbolic partial differentialudequations, for which the exact solution is usually not known. Numerical methodsudare employed to approximate the solution to hyperbolic problems; however, in manyudcases it is difficult to satisfy certain physical properties while maintaining high orderudof accuracy. In this thesis, we develop high-order time-stepping methods thatudare capable of maintaining stability constraints of the solution, when coupled withudsuitable spatial discretizations. Such methods are called strong stability preservingud(SSP) time integrators, and we mainly focus on perturbed methods that use bothudupwind- and downwind-biased spatial discretizations.udFirstly, we introduce a new family of third-order implicit Runge–Kuttas methodsudwith arbitrarily large SSP coefficient. We investigate the stability and accuracy ofudthese methods and we show that they perform well on hyperbolic problems with largeudCFL numbers. Moreover, we extend the analysis of SSP linear multistep methods toudsemi-discretized problems for which different terms on the right-hand side of theudinitial value problem satisfy different forward Euler (or circle) conditions. Optimaludperturbed and additive monotonicity-preserving linear multistep methods are studiedudin the context of such problems. Optimal perturbed methods attain augmentedudmonotonicity-preserving step sizes when the different forward Euler conditions areudtaken 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 correspondingudnon-additive SSP linear multistep methods. Furthermore, we develop theudfirst SSP linear multistep methods of order two and three with variable step size, andudstudy their optimality. We describe an optimal step-size strategy and demonstrateudthe effectiveness of these methods on various one- and multi-dimensional problems.udFinally, we establish necessary conditions to preserve the total variation of the solutionudobtained when perturbed methods are applied to boundary value problems.udWe implement a stable treatment of nonreflecting boundary conditions for hyperbolicudproblems that allows high order of accuracy and controls spurious wave reflections.udNumerical examples with high-order perturbed Runge–Kutta methods reveal that thisudtechnique provides a significant improvement in accuracy compared with zero-orderudextrapolation.
展开▼
