Spatially partitioned embedded Runge-Kutta Methods

We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory

A conservative implicit multirate method for hyperbolic problems

This work focuses on the development of a self adjusting multirate strategy based on an implicit time discretization for the numerical solution of hyperbolic equations, that could benefit from different time steps in different areas of the spatial domain. We propose a novel mass conservative multirate approach, that can be generalized to various implicit time discretization methods. It is based on flux partitioning, so that flux exchanges between a cell and its neighbors are balanced. A number of numerical experiments on both non-linear scalar problems and systems of hyperbolic equations have been carried out to test the efficiency and accuracy of the proposed approach

Adaptive Multirate Infinitesimal Time Integration

As multiphysics simulations grow in complexity and application scientists desire more accurate results, computational costs increase greatly. Time integrators typically cater to the most restrictive physical processes of a given simulation\add{,} which can be unnecessarily computationally expensive for the less restrictive physical processes. Multirate time integrators are a way to combat this increase in computational costs by efficiently solving systems of ordinary differential equations that contain physical processes which evolve at different rates by assigning different time step sizes to the different processes. Adaptivity is a technique for further increasing efficiency in time integration by automatically growing and shrinking the time step size to be as large as possible to achieve a solution accurate to a prescribed tolerance value. Adaptivity requires a time step controller, an algorithm by which the time step size is changed between steps, and benefits from an integrator with an embedding, an efficient way of estimating the error arising from each step of the integrator. In this thesis, we develop these required aspects for multirate infinitesimal time integrators, a subclass of multirate time integrators which allow for great flexibility in the treatment of the processes that evolve at the fastest rates. First, we derive the first adaptivity controllers designed specifically for multirate infinitesimal methods, and we discuss aspects of their computational implementation. Then, we derive a new class of efficient, flexible multirate infinitesimal time integrators which we name implicit-explicit multirate infinitesimal stage-restart (IMEX-MRI-SR) methods. We derive conditions guaranteeing up to fourth-order accuracy of IMEX-MRI-SR methods, explore their stability properties, provide example methods of orders two through four, and discuss their performance. Finally, we derive new instances of the class of implicit-explicit multirate infinitesimal generalized-structure additive Runge-Kutta methods, developed by Chinomona and Reynolds (2022), with embeddings and explore their stability properties and performance