1,721,034 research outputs found
Strong stability preserving implicit–explicit transformed general linear methods
No full textWe consider the class of implicit–explicit (IMEX) general linear methods (GLMs) to construct methods where the explicit part has strong stability preserving (SSP) property, while the implicit part of the method has inherent Runge–Kutta stability (IRKS) property, and it is A-, or L-stable. We will also investigate the absolute stability of these methods when the implicit and explicit parts interact with each other. In particular, we will monitor the size of the region of absolute stability of the IMEX scheme, assuming that the implicit part of the method is A(α)-stable for α∈[0,π∕2]. Finally we furnish examples of SSP IMEX GLMs up to the order p=4 and stage order q=p with optimal SSP coefficients
General Linear Methods with external stages of different orders
No full textThe approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. in this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna
Construction of SDIRK methods with dispersive stability functions
No full textWe describe a new approach to the construction of rational functions of high dispersive orders, and SDIRK methods with dispersive stability functions, for the numerical solution of differential systems with oscillatory solutions. The numerical experiments on test problems with periodic or almost periodic solutions confirm the order and dispersive order of convergence of the proposed numerical schemes
Order conditions for partitioned Runge-Kutta Methods
No full textWe illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations
Strong Stability Preserving Runge–Kutta and Linear Multistep Methods
No full textThis paper reviews strong stability preserving discrete variable methods for differential systems. The strong stability preserving Runge–Kutta methods have been usually investigated in the literature on the subject, using the so-called Shu–Osher representation of these methods, as a convex combination of first-order steps by forward Euler method. In this paper, we revisit the analysis of strong stability preserving Runge–Kutta methods by reformulating these methods as a subclass of general linear methods for ordinary differential equations, and then using a characterization of monotone general linear methods, which was derived by Spijker in his seminal paper (SIAM J Numer Anal 45:1226–1245, 2007). Using this new approach, explicit and implicit strong stability preserving Runge–Kutta methods up to the order four are derived. These methods are equivalent to explicit and implicit RK methods obtained using Shu–Osher or generalized Shu–Osher representation. We also investigate strong stability preserving linear multistep methods using again monotonicity theory of Spijker
- …
