arXiv subjects

Daniel R. Reynolds

Publications and source records attributed to Daniel R. Reynolds.

At least 19 recordsLinked to original sources

arXiv2026

Implicit-explicit and split-explicit super-time-stepping methods

Multiphysics initial-value problems couple processes with distinct stability properties, such as advection, diffusion, and stiff local reactions. Standard implicit-explicit (ImEx) additive Runge--Kutta (ARK) methods can treat these processes accurately, but require globally coupled implicit solves when diffusion is grouped with reaction; operator splitting avoids such solves but typically provides weaker coupling and no inexpensive temporal error estimate; and PIROCK is tied to a specific Runge--Kutta--Chebyshev super-time-stepping (STS) construction. We introduce extended super-time-stepping (ExtSTS) methods, a family of time integration schemes that combine super-time-stepping methods for diffusive terms with explicit, implicit, or ImEx Runge--Kutta treatment of the remaining terms. The coupling is based on multirate infinitesimal techniques, yielding solve-decoupled methods that retain localized implicit solves, support embedded error estimation for adaptive time stepping, and allow flexible use of modern STS methods. We present the ExtSTS method family, provide a robust technique for ExtSTS method creation, formulate the corresponding linear stability theory, and construct embedded ImEx, explicit, and implicit ExtSTS methods. Numerical experiments on one- and two-dimensional advection-diffusion-reaction problems show that ExtSTS methods are robust across parameter regimes and boundary conditions, and are often more efficient than ARK, Strang splitting, and PIROCK methods, especially when strong coupling between operators is important.

math.NA↗

arXiv2026

Efficient and Flexible Multirate Temporal Adaptivity

In this work we present two new families of multirate time step adaptivity controllers, that are designed to work with embedded multirate infinitesimal (MRI) time integration methods for adapting time steps when solving problems with multiple time scales. We compare these controllers against competing approaches on two benchmark problems, showing that the proposed methods offer dramatically improved performance and flexibility. The combination of embedded MRI methods and the proposed controllers enable adaptive simulations of problems with a potentially arbitrary number of time scales, achieving high accuracy while maintaining low computational cost. Additionally, we introduce a new set of embeddings for the family of explicit multirate exponential Runge--Kutta (MERK) methods of orders 2 through 5, resulting in the first-ever fifth-order embedded MRI method. Finally, we compare the performance of a wide range of embedded MRI methods on our benchmark problems to provide guidance on how to select an appropriate MRI method and multirate controller.

math.NA↗

arXiv2026

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↗

arXiv2026

Super Time Stepping Methods for Diffusion using Discontinuous-Galerkin Spatial Discretizations

Super-time-stepping (STS) methods provide an attractive approach for enabling explicit time integration of parabolic operators, particularly in large-scale, higher-dimensional kinetic simulations where fully implicit schemes are impractical. In this work, we present an explicit STS framework tailored for diffusion operators in gyrokinetic models, motivated by the fact that constructing and storing a Jacobian is often infeasible due to strong nonlocal couplings, high dimensionality, and memory constraints. We investigate the performance of several STS methods, including Runge-Kutta-Chebyshev (RKC) and Runge-Kutta-Legendre (RKL) schemes, applied to a diffusion equation discretized using both discontinuous Galerkin (DG) and finite-difference methods. To support time adaptivity, we introduce a novel error norm designed to more accurately track temporal error arising from DG spatial discretizations, in which degrees of freedom contribute unevenly to the solution error. Finally, we assess the performance of an automatic eigenvalue estimation algorithm for determining the required number of STS stages and compare it against an analytical estimation formula.

math.NA↗

arXiv2024

SUNDIALS Time Integrators for Exascale Applications with Many Independent ODE Systems

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world's largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This operator splitting approach is used ubiquitously across scientific domains, and in many cases leads to a set of ordinary differential equations (ODEs) that need to be solved as part of a larger "outer-loop" time-stepping approach. The SUNDIALS library provides a plethora of robust time integration algorithms for solving ODEs, and the U.S. Department of Energy Exascale Computing Project (ECP) has supported its extension to applications on exascale-capable computing hardware. In this paper, we highlight some SUNDIALS capabilities and its deployment in combustion and cosmology application codes (Pele and Nyx, respectively) where operator splitting gives rise to numerous, small ODE systems that must be solved concurrently.

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↗

arXiv2022

ARKODE: a flexible IVP solver infrastructure for one-step methods

We describe the ARKODE library of one-step time integration methods for ordinary differential equation (ODE) initial-value problems (IVPs). In addition to providing standard explicit and diagonally implicit Runge--Kutta methods, ARKODE also supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge--Kutta methods and multirate infinitesimal (MRI) methods. We present the role of ARKODE within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKODE infrastructure for utilities common to large classes of one-step methods, as well as its use of ``time stepper'' modules enabling easy incorporation of novel algorithms into the library. Numerical results show example problems of increasing complexity, highlighting the algorithmic flexibility afforded through this infrastructure, and include a larger multiphysics application leveraging multiple algorithmic features from ARKODE and SUNDIALS.

cs.MS↗

arXiv2022

Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks

Large-scale multiphysics simulations are computationally challenging due to the coupling of multiple processes with widely disparate time scales. The advent of exascale computing systems exacerbates these challenges, since these enable ever increasing size and complexity. Recently, there has been renewed interest in developing multirate methods as a means to handle the large range of time scales, as these methods may afford greater accuracy and efficiency than more traditional approaches of using IMEX and low-order operator splitting schemes. However, there have been few performance studies that compare different classes of multirate integrators on complex application problems. We study the performance of several newly developed multirate infinitesimal (MRI) methods, implemented in the SUNDIALS solver package, on two reacting flow model problems built on structured mesh frameworks. The first model revisits the work of Emmet et al. (2014) on a compressible reacting flow problem with complex chemistry that is implemented using BoxLib but where we now include comparisons between a new explicit MRI scheme with the multirate spectral deferred correction (SDC) methods in the original paper. The second problem uses the same complex chemistry as the first problem, combined with a simplified flow model, but run at a large spatial scale where explicit methods become infeasible due to stability constraints. Two recently developed implicit-explicit MRI multirate methods are tested. These methods rely on advanced features of the AMReX framework on which the model is built, such as multilevel grids and multilevel preconditioners. The results from these two problems show that MRI multirate methods can offer significant performance benefits on complex multiphysics application problems and that these methods may be combined with advanced spatial discretization to compound the advantages of both.

math.NA↗

arXiv2022

Adaptive time step control for multirate infinitesimal methods

Multirate methods have been used for decades to temporally evolve initial-value problems in which different components evolve on distinct time scales, and thus use of different step sizes for these components can result in increased computational efficiency. Generally, such methods select these different step sizes based on experimentation or stability considerations. For problems that evolve on a single time scale, adaptivity approaches that strive to control local temporal error are widely used to achieve numerical results of a desired accuracy with minimal computational effort, while alleviating the need for manual experimentation with different time step sizes. However, there is a notable gap in the publication record on the development of adaptive time-step controllers for multirate methods. In this paper, we extend the single-rate controller work of Gustafsson (1994) to the multirate method setting. Specifically, we develop controllers based on polynomial approximations to the principal error functions for both the "fast" and "slow" time scales within multirate infinitesimal (MRI) methods. We additionally investigate a variety of approaches for estimating the errors arising from each time scale within MRI methods. We then numerically evaluate the proposed multirate controllers and error estimation strategies on a range of multirate test problems, comparing their performance against an estimated optimal performance. Through this work, we combine the most performant of these approaches to arrive at a set of multirate adaptive time step controllers that robustly achieve desired solution accuracy with minimal computational effort.

math.NA↗

arXiv2021

Enabling GPU Accelerated Computing in the SUNDIALS Time Integration Library

As part of the Exascale Computing Project (ECP), a recent focus of development efforts for the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been to enable GPU-accelerated time integration in scientific applications at extreme scales. This effort has resulted in several new GPU-enabled implementations of core SUNDIALS data structures, support for programming paradigms which are aware of the heterogeneous architectures, and the introduction of utilities to provide new points of flexibility. In this paper, we discuss our considerations, both internal and external, when designing these new features and present the features themselves. We also present performance results for several of the features on the Summit supercomputer and early access hardware for the Frontier supercomputer, which demonstrate negligible performance overhead resulting from the additional infrastructure and significant speedups when using both NVIDIA and AMD GPUs.

cs.DC↗

arXiv2021

Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers

In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through the addition of new linear and nonlinear solvers classes, enhancements to the vector class, and the creation of modern Fortran interfaces. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. These changes allow SUNDIALS users to more easily utilize external solver libraries and create highly customized solvers, enabling greater flexibility on extreme-scale, heterogeneous computational architectures.

cs.MS↗

arXiv2021

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix $φ$-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods.

math.NA↗

arXiv2021

Implicit-explicit multirate infinitesimal GARK methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

math.NA↗

arXiv2019

Evaluation of Implicit-Explicit Additive Runge-Kutta Integrators for the HOMME-NH Dynamical Core

The nonhydrostatic High Order Method Modeling Environment (HOMME-NH) atmospheric dynamical core supports acoustic waves that propagate significantly faster than the advective wind speed, thus greatly limiting the timestep size that can be used with standard explicit time-integration methods. Resolving acoustic waves is unnecessary for accurate climate and weather prediction. This numerical stiffness is addressed herein by considering implicit-explicit additive Runge-Kutta (ARK IMEX) methods that can treat the acoustic waves in a stable manner without requiring implicit treatment of non-stiff modes. Various ARK IMEX methods are evaluated for their efficiency in producing accurate solutions, ability to take large timestep sizes, and sensitivity to grid cell length ratio. Both the Gravity Wave test and Baroclinic Instability test from the 2012 Dynamical Core Model Intercomparison Project (DCMIP) are used to recommend 5 of the 27 ARK IMEX methods for use in HOMME-NH.

math.NA↗

arXiv2019

SUNDIALS Multiphysics+MPIManyVector Performance Testing

In this report we document performance test results on a SUNDIALS-based multiphysics demonstration application. We aim to assess the large-scale parallel performance of new capabilities that have been added to the SUNDIALS suite of time integrators and nonlinear solvers in recent years under funding from both the Exascale Computing Project (ECP) and the Scientific Discovery through Advanced Scientific (SciDAC) program, specifically: (a) SUNDIALS' new MPIManyVector module, that allows extreme flexibility in how a solution "vector" is staged on computational resources, (b) ARKode's new multirate integration module, MRIStep, allowing high-order accurate calculations that subcycle "fast" processes within "slow" ones, (c) SUNDIALS' new flexible linear solver interfaces, that allow streamlined specification of problem-specific linear solvers, and (d) SUNDIALS' new N_Vector additions of "fused" vector operations (to increase arithmetic intensity) and separation of reduction operations into "local" and "global" versions (to reduce latency by combining multiple reductions into a single MPI_Allreduce call). We anticipate that subsequent reports will extend this work to investigate a variety of other new features, including SUNDIALS' generic SUNNonlinearSolver interface and accelerator-enabled N_Vector modules, and upcoming MRIStep extensions to support custom "fast" integrators (that leverage problem structure) and IMEX integration of the "slow" time scale (to add diffusion).

cs.DC↗

arXiv2019

Relaxed Multirate Infinitesimal Step Methods for Initial-Value Problems

This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate (RFSMR) version of the Multirate Infinitesimal Step (MIS) methods and use recent theoretical developments for Generalized Additive Runge-Kutta methods to propose our higher-order Relaxed Multirate Infinitesimal Step extensions. The resulting framework supports a range of attractive properties for multirate methods, including telescopic extensions, subcycling, embeddings for temporal error estimation, and support for changes to the fast/slow time-scale separation between steps, without requiring any sacrifices in linear stability. In addition to providing rigorous theoretical developments for these new methods, we provide numerical tests demonstrating convergence and efficiency on a suite of multirate test problems.

math.NA↗

arXiv2019

A new class of high-order methods for multirate differential equations

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified "fast" initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these Multirate Exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order $p$, the fast problems corresponding to internal stages may be solved using a method of order $p-1$, while the final fast problem corresponding to the time-evolved solution must use a method of order $p$. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.

math.NA↗

arXiv2018

Further development of efficient and accurate time integration schemes for meteorological models

In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods on the shallow water equations on the sphere. This stiff, nonlinear model provides a testing ground for accurate and stable time integration methods in weather modeling, serving as the focus for exploration of novel methods for many years. We therefore identify a candidate set of three recent exponential Rosenbrock methods of orders four and five (exprb42, pexprb43 and exprb53) for use on this model. Based on their multi-stage structure, we propose a set of modifications to the phipm_IOM2 algorithm for efficiently calculating the matrix phi-functions. We then investigate the performance of these methods on a suite of four challenging test problems, comparing them against the epi3 method investigated previously in [1, 2] on these problems. In all cases, the proposed methods enable accurate solutions at much longer time-steps than epi3, proving considerably more efficient as either the desired solution error decreases, or as the test problem nonlinearity increases.

math.NA↗