Peer Methods for the Solution of Large-Scale Differential Matrix Equations
For computational scientists solving large-scale matrix differential equations, this work offers more efficient time integration methods, though the improvement is incremental over existing approaches.
The paper applies implicit and Rosenbrock-type peer methods to large-scale differential Riccati equations, developing a reformulation to reduce computational complexity and a low-rank factorization implementation. Numerical experiments show peer methods up to order 4 outperform existing implicit schemes in efficiency.
We consider the application of implicit and linearly implicit (Rosenbrock-type) peer methods to matrix-valued ordinary differential equations. In particular the differential Riccati equation (DRE) is investigated. For the Rosenbrock-type schemes, a reformulation capable of avoiding a number of Jacobian applications is developed that, in the autonomous case, reduces the computational complexity of the algorithms. Dealing with large-scale problems, an efficient implementation based on low-rank symmetric indefinite factorizations is presented. The performance of both peer approaches up to order 4 is compared to existing implicit time integration schemes for matrix-valued differential equations.