arXiv subjects

Steven B. Roberts

Publications and source records attributed to Steven B. Roberts.

6 recordsLinked to original sources

arXiv2026

Adaptive Diagonally Implicit Runge-Kutta Methods Devoid of Order Reduction for Semilinear ODEs

Diagonally implicit Runge-Kutta (DIRK) methods are a prominent class of numerical methods for solving stiff systems of ordinary differential equations (ODEs). Stiffness does not only impose stability challenges on Runge-Kutta methods; it can also degrade the order of convergence. This so-called order reduction phenomenon occurs when assumptions used for classical convergence analysis, e.g., an asymptotically small step size, fail to hold. In a prior paper by the authors, sharp order conditions and global error bounds for Runge-Kutta methods were developed, which hold uniformly with respect to stiffness when applied to a wide class of semilinear ODEs. In this work, those conditions are leveraged to construct the first DIRK methods of order four and five which satisfy these conditions and thus do not exhibit order reduction. Numerical results demonstrate that for a broad class of relevant nonlinear test problems, these new methods successfully mitigate order reduction, accurately estimate local error via an embedding for adaptive step size control, and can outperform classical DIRK methods.

math.NA↗

arXiv2026

Leveraging higher-order time integration methods for improved computational efficiency in a rainshaft model

Cloud and precipitation microphysics packages in atmospheric general circulation models typically use first-order time integration methods with a large time step, requiring ad hoc limiters and substepping of the sedimentation scheme to prevent solutions from becoming unstable. We show that in the latest version of Energy Exascale Earth System Model, E3SMv3, the rain microphysics provided by the Predicted Particle Properties (P3) scheme is underresolved in time at the model's default 300s time step. The P3 scheme requires limiters to guarantee stability, but those limiters make large discretization errors more difficult to detect. When the time step of the P3 scheme is reduced to sufficiently capture correct microphysics behavior, wall clock time of the simulation is increased by nearly a factor of 40. Instead of reducing the microphysics time step, we recommend using higher-order time integrators based on Runge-Kutta methods, which offer improved solution accuracy at comparable computational costs. A key to obtaining computationally efficient microphysics results is the use of adaptive time stepping, which also eliminates the need for specialized substepping procedures in the sedimentation process. We also analyze individual microphysical processes by extracting inverse timescales from Jacobians of the process rates, which gives insight about the maximum time step each process is able to take while maintaining stability and accuracy, and about how individual processes should be grouped together for most efficient results. The proposed integrators can achieve the accuracy level required to correctly model rain microphysics parameterizations more than 10x faster than the P3 scheme.

physics.ao-ph↗

arXiv2025

New Time Integrators and Capabilities in SUNDIALS Versions 6.2.0-7.4.0

SUNDIALS is a well-established numerical library that provides robust and efficient time integrators and nonlinear solvers. This paper overviews several significant improvements and new features added over the last three years to support scientific simulations run on high-performance computing systems. Notably, three new classes of one-step methods have been implemented: low storage Runge-Kutta, symplectic partitioned Runge-Kutta, and operator splitting. In addition, we describe new time step adaptivity support for multirate methods, adjoint sensitivity analysis capabilities for explicit Runge-Kutta methods, additional options for Anderson acceleration in nonlinear solvers, and improved error handling and logging.

cs.MS↗

arXiv2025

A Stiff Order Condition Theory for Runge-Kutta Methods Applied to Semilinear ODEs

Classical convergence theory of Runge-Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss-Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term.

math.NA↗

arXiv2025

Moment-based adaptive time integration for thermal radiation transport

In this paper we develop a framework for moment-based adaptive time integration of deterministic multifrequency thermal radiation transpot (TRT). We generalize our recent semi-implicit-explicit (IMEX) integration framework for gray TRT to multifrequency TRT, and also introduce a semi-implicit variation that facilitates higher-order integration of TRT, where each stage is implicit in all components except opacities. To appeal to the broad literature on adaptivity with Runge--Kutta methods, we derive new embedded methods for four asymptotic preserving IMEX Runge--Kutta schemes we have found to be robust in our previous work on TRT and radiation hydrodynamics. We then use a moment-based high-order-low-order representation of the transport equations. Due to the high dimensionality, memory is always a concern in simulating TRT. We form error estimates and adaptivity in time purely based on temperature and radiation energy, for a trivial overhead in computational cost and memory usage compared with the base second order integrators. We then test the adaptivity in time on the tophat and Larsen problem, demonstrating the ability of the adaptive algorithm to naturally vary the timestep across 4--5 orders of magnitude, ranging from the dynamical timescales of the streaming regime to the thick diffusion limit.

math.NA↗

arXiv2023

Implicit-Explicit Multirate Infinitesimal Stage-Restart Methods

Implicit-Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is partitioned into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems partitioned into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of ``fast'' IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit-explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further partitioned in an IMEX fashion. This new class, which we call implicit-explicit multirate stage-restart (IMEX-MRI-SR), both improves upon the previous implicit-explicit multirate generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods, and extends multirate exponential Runge Kutta (MERK) methods into the IMEX context. We leverage GARK theory to derive conditions guaranteeing orders of accuracy up to four. We provide second-, third-, and fourth-order accurate example methods and perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

math.NA↗