CUDA acceleration of a matrix-free Rosenbrock-K method applied to the shallow water equations

Many simulations of evolutionary ordinary and partial differential equations require implicit time integration methods to avoid stability restrictions on the step size. The computation and communication costs associated with solving nonlinear systems at each time step dominates the total simulation cost. Rosenbrock-Krylov (Rosenbrock-K) methods alleviate this major bottleneck by using Krylov space approximations tightly coupled with the time discretization. This work studies the performance of Rosenbrock-K methods on accelerated hardware. GPU acceleration is used to expedite computations of the semi-discrete right hand side and the linear-algebra computations in the time integration method. A novel parallelization of the Arnoldi procedure for the construction of the Krylov based approximations of the Jacobian matrix is presented. Rosenbrock-K methods' unique ability to operate almost entirely in a reduced space make them especially suitable for efficient utilization of accelerated hardware, where standard implicit approaches may lead to systems too large for device memory.