Abstract
Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.
Highlights
Based on the fact that integration by parts plays a major role in the development of energy and entropy estimates for initial boundary value problems, one may conjecture that the summation by parts (SBP) property [8,47] is a key factor in provably stable schemes
Having the conjecture “stability results require an SBP structure” in mind, this article provides additional insights to this topic by constructing new classes of SBP schemes, which reduce to the classical Lobatto IIIA and Lobatto IIIB methods if that quadrature rule is used
All A stable classical Runge-Kutta methods based on Radau and Lobatto quadrature rules can be formulated in the framework of SBP operators
Summary
Based on the fact that integration by parts plays a major role in the development of energy and entropy estimates for initial boundary value problems, one may conjecture that the summation by parts (SBP) property [8,47] is a key factor in provably stable schemes. Turning to SBP methods in time [2,19,25], a class of linearly and nonlinearly stable SBP schemes has been constructed and studied in this context, see [18,41,42]. If the underlying quadrature is chosen as Radau or Lobatto quadrature, these Runge-Kutta schemes are exactly the classical Radau IA, Radau IIA, and Lobatto IIIC methods [29]. Having the conjecture “stability results require an SBP structure” in mind, this article provides additional insights to this topic by constructing new classes of SBP schemes, which reduce to the classical Lobatto IIIA and Lobatto IIIB methods if that quadrature rule is used. All A stable classical Runge-Kutta methods based on Radau and Lobatto quadrature rules can be formulated in the framework of SBP operators. Instead of using simultaneous approximation terms (SATs) [5,6] to impose initial conditions weakly, these new schemes use a strong imposition of initial conditions in combination with a projection method [22,26,27]
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