Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Stability and convergence at the PDE/stiff ODE interface
Applied Numerical Mathematics - Recent Theoretical Results in Numerical Ordinary Differential Equations
SIAM Journal on Scientific Computing
On Strong Stability Preserving Time Discretization Methods
Journal of Scientific Computing
High-order linear multistep methods with general monotonicity and boundedness properties
Journal of Computational Physics
On High Order Strong Stability Preserving Runge---Kutta and Multi Step Time Discretizations
Journal of Scientific Computing
General linear methods for ordinary differential equations
Mathematics and Computers in Simulation
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This paper introduces a three and a four order explicit time stepping method. These methods have high stage order and favorable monotonicity properties. The proposed methods are based on multistage-multistep (MM) schemes that belong to the broader class of general linear methods, which are generalizations of both Runge-Kutta and linear multistep methods. Methods with high stage order alleviate the order reduction occurring in explicit multistage methods due to non-homogeneous boundary/source terms. Furthermore, the MM schemes presented in this paper can be expressed as convex combinations of Euler steps. Consequently, they have the same monotonicity properties as the forward Euler method. This property makes these schemes well suited for problems with discontinuous solutions.