Rodas6P and Tsit5DA - two new Rosenbrock-type methods for DAEs
Gerd Steinebach
TL;DR
The paper addresses efficient numerical solution of index-1 DAEs by expanding the Rosenbrock family with two new methods: Rodas6P, a sixth-order stiffly accurate Rosenbrock-type method with embedded fifth order, and Tsit5DA, a fifth-order method that blends an explicit differential solver with a linear-implicit algebraic component based on Tsit5. Rodas6P achieves high accuracy at the cost of many stages (s=19) but offers potential time savings for large, semi-discretized PDE-DAEs and remains L-stable; Tsit5DADA provides a significant improvement over RKF4DA for non-stiff problems by leveraging an explicit-implicit split and optimized coefficients. The authors validate theoretical orders via order tests and benchmarks, including Prothero-Robinson-type problems, semi-discretized parabolic and hyperbolic PDEs, and a multi-pendulum example, and supply extensive coefficient data and implementation details. Together, Rodas6P and Tsit5DA extend the Rosenbrock family with a high-order stiff method and a fifth-order explicit-implicit DAE solver, offering practitioners options for balancing accuracy, stiffness handling, and computational efficiency in large-scale DAEs.
Abstract
Two new Rosenbrock methods for solving index-1 differential algebraic equations are presented. Rodas6P is a sixth-order method based on the same design principles as the Rodas3P, Rodas4P, and Rodas5P methods. Tsit5DA is based on an explicit solution approach for the differential equations and a linear-implicit approach for the algebraic equations. Such a fourth-order method has already been presented in Rentrop, Roche & Steinebach, 1989. Tsit5DA now provides a significantly improved fifth-order method which is based on the well known Tsit5 method. The theoretical properties of the new methods are verified by some order tests and benchmarks.