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Organizing Committee
University of Massachusetts Dartmouth
University of Massachusetts Dartmouth
University of Washington
King Abdullah University of Science & Technology
Abstract
A universal source of difficulty in the numerical solution of differential equations is stiffness, which may stem from multi-scale dynamics and means that straightforward explicit discretizations are very inefficient. Consequently, choosing how to deal with stiffness (usually through the selection of the time discretization method) is often the most impactful decision made while solving these problems; it can be the difference between complete intractability and rapid solution.
The most prominent and well-known effect of stiffness is its impact on numerical stability. Lesser-known but often equally important is the effect of order reduction, in which the convergence of the numerical solution is dramatically slowed compared to what classical numerical analysis would predict. Both of these issues can be dealt with through the use of implicit time discretizations. However, for large-scale applications (such as those coming from multi-dimensional PDE applications), such discretizations can entail an even higher cost as they require the solution of large systems of algebraic equations. Ongoing development in the field of time discretization now focuses on methods that are increasingly specialized and, in some cases, increasingly complex. The aim of this workshop is to advance the field of time evolution for PDEs by bringing together a broad community that will address this important issue in a variety of ways.
The most prominent and well-known effect of stiffness is its impact on numerical stability. Lesser-known but often equally important is the effect of order reduction, in which the convergence of the numerical solution is dramatically slowed compared to what classical numerical analysis would predict. Both of these issues can be dealt with through the use of implicit time discretizations. However, for large-scale applications (such as those coming from multi-dimensional PDE applications), such discretizations can entail an even higher cost as they require the solution of large systems of algebraic equations. Ongoing development in the field of time discretization now focuses on methods that are increasingly specialized and, in some cases, increasingly complex. The aim of this workshop is to advance the field of time evolution for PDEs by bringing together a broad community that will address this important issue in a variety of ways.
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Workshop Schedule
Monday, July 21, 2025
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
Mixed model and mixed precision Runge--Kutta methods
11th Floor Lecture Hall
Speaker
Sigal Gottlieb, University of Massachusetts Dartmouth
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
In this talk we present a mixed precision and mixed model approach for multi-stage methods. We show that Runge-Kutta methods can be designed so that certain costly intermediate computations can be performed as a lower-cost computation without adversely impacting the accuracy of the overall solution. In particular, a properly designed Runge-Kutta method will damp out the errors committed in the initial stages. This is of particular interest when we consider implicit Runge-Kutta methods. In such cases, the implicit computation of the stage values can be considerably faster if the solution can be of lower precision, lower tolerance, or cheaper model. We provide a general theoretical additive framework for designing mixed precision Runge-Kutta methods, and use this framework to derive order conditions for such methods. Next, we show how using this approach allows us to leverage cheaper computation of the implicit solver while retaining higher accuracy in the overall method. Finally, we perform a stability analysis which informs how to design methods that will be stable for different types of splittings to determine the cheaper but less accurate model.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
IMplicit EXplicit Runge-Kutta methods for stiff problems
11th Floor Lecture Hall
Speaker
Inmaculada Higueras, Public University of Navarre
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
Spatial discretization of some partial differential equations (PDEs) gives rise to stiff systems of ordinary differential equations (ODEs). This is the case, for example, for equations of motion associated with the semi-convection problem in the simulation of stellar convection, and for some other models of flow with diffusion that appear in astrophysics.
To solve the system of ODEs, IMplicit-EXplicit (IMEX) time stepping methods, specially constructed to have different stability properties, are used. Ongoing work continues with the study of some implementation issues.
11:30 AM - 12:15 PM EDT
Uniform accuracy of implicit-explicit methods for stiff hyperbolic relaxation systems and kinetic equations
11th Floor Lecture Hall
Speaker
Ruiwen Shu, University of Georgia
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) methods have been widely used and their performance is understood well in the non-stiff regime ($\varepsilon=O(1)$) and limiting regime ($\varepsilon\rightarrow 0$). However, in the intermediate regime (say, $\varepsilon=O(\Delta t)$), uniform accuracy has been reported numerically for most IMEX multistep methods, while complicated behavior of order reduction has been observed for IMEX Runge-Kutta (RK) methods. In this talk, I will take a linear hyperbolic systems with stiff relaxation as a model problem, and discuss how to use energy estimates with multiplier techniques to prove the uniform accuracy of IMEX methods. In particular, I will present my joint works with Jingwei Hu on the uniform accuracy of IMEX backward differentiation formulas (IMEX-BDF) up to fourth order, and that of IMEX-RK methods up to third order.
12:30 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
Integrating hyperbolic conservation laws with high order implicit schemes
11th Floor Lecture Hall
Speaker
Gabriella Puppo, La Sapienza Università di Roma.
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
Systems of hyperbolic conservation laws often present different scales linked to
the magnitude of the eigenvalues of the Jacobian of the flux function. Numerical schemes used to integrate hyperbolic problems usually are explicit, because in many applications one is interested in resolving all scales. Borrowing our terminology from gas dynamics, explicit schemes resolve both convective and acoustic waves.
However, when the flow is considerably slower than the sound speed, the stability condition required by an explicit scheme may become too restrictive. Moreover, the solution obtained resolves the fast scales far better than the slow ones. Thus, an implicit scheme may yield not only a more favourable computational time, but also a sharper resolution of the slow scales which may be of interest. This occurs for instance when one is more interested in the evolution of vortices rather than the propagation of the corresponding sound.
In this contribution, we will focus on a class of implicit schemes designed with this purpose. The main point is to base the computation on a low order predictor, coupled with a higher order corrector. The resulting scheme provides a non oscillatory solution, degrading to first order as the scheme detects the presence of shock waves.
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:15 PM EDT
Strong stability two derivative Runge--Kutta methods
11th Floor Lecture Hall
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the strong stability preserving theory for explicit and implicit two-derivative Runge–Kutta schemes, including a special condition on the second derivative under which these implicit methods may be unconditionally strong stability preserving. This special condition is natural for the stiff component of wide range of plasma physics problems, and can be useful in the context of strong stability preserving implicit-explicit multi-derivative Runge–Kutta schemes, where the time-step restriction is then independent of the stiff term. Finally, we present the strong stability preserving theory for implicit-explicit multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.
This is a joint work with Andrew Christlieb, Sining Gong, Sigal Gottlieb, Jingwei Hu, and Ruiwen Shu.
4:30 - 5:00 PM EDT
Types of Time Integrators
Panel Discussion - 11th Floor Lecture Hall
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Panelist
Sebastiano Boscarino, University of Catania
Panelist
Inmaculada Higueras, Public University of Navarre
Panelist
Jingwei Hu, University of Washington
Panelist
Mayya Tokman, University of California, Merced
Abstract
We will discuss experiences with IMEX, splitting methods, exponential methods, etc for different problems
to convey a sense of which methods work well in what settings
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, July 22, 2025
9:30 - 10:00 AM EDT
Coffee Break
Virtual
10:00 - 10:15 AM EDT
Poster Blitz
Lightning Talks - 11th Floor Lecture Hall
Speaker
Fred Daum, Raytheon
Speaker
Cesar Herrera Garro, Purdue University
Speaker
Saburi Rasheed, University of Louisiana at Lafayette
Speaker
John Stephens, Baylor University
Speaker
Shiheng Zhang, University of Washington
Session Chair
Jingwei Hu, University of Washington
10:20 AM - 12:00 PM EDT
12:00 - 1:30 PM EDT
Networking Lunch
Working Lunch
1:30 - 2:15 PM EDT
Constructing and Using Exponential Integrators for Stiff Systems.
11th Floor Lecture Hall
Speaker
Mayya Tokman, University of California, Merced
Session Chair
Jingwei Hu, University of Washington
Abstract
Over the past several decades exponential integration has become a more prominent topic in numerical analysis and scientific computing. It is still, however, a new-kid-on-the-block compared to the extensively studied and used explicit and implicit temporal integrators. Thus, no broad recognition exists in the scientific community on how to answer questions like when one should use an exponential method, what kind of exponential schemes are appropriate for specific problems or what an efficient implementation of an exponential integrator looks like. In this talk, we will attempt to address some of these issues and present considerations that can guide the construction, analysis and application of exponential methods. We will discuss how different classes of exponential integrators can be derived and outline the intricacies of their implementation particularly for high performance computing platforms. A novel approach to derivation of exponential schemes using stiffness-resilient framework will be described and performance of the novel methods will be demonstrated on a suite of numerical tests. We will show how taking into account the structure of the application of interest can lead to construction of exponential schemes particularly efficient for this problem. Examples from fields such as plasma physics, numerical weather prediction and computer graphics will be used.
2:30 - 3:15 PM EDT
On the accuracy and stability of BDF schemes from an SBP perspective
11th Floor Lecture Hall
Speaker
Ayaboe Edoh, Amentum
Session Chair
Jingwei Hu, University of Washington
Abstract
The current work considers the accuracy and provable stability of multi-step backward differencing formula (BDF) methods. The BDF progression is expressed as a summation-by-parts (SBP) operator in time, for which an associated quadrature ensues. This in turn provides a convenient means of assessing final-time accuracy and yields a natural norm for analyzing discrete energy stability. The proposed framework confirms the utility of employing alternating BDF formulas (e.g., mixing BDF2 and BDF3 steps) and provides a new way of characterizing the robustness and accuracy of adaptive BDF procedures for stiff systems.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Localized Exponential Time Differencing Methods for Evolution Partial Differential Equations
11th Floor Lecture Hall
Speaker
Thi Thao Phuong Hoang, Auburn University
Session Chair
Jingwei Hu, University of Washington
Abstract
Exponential integrators, among them the Exponential Time Differencing (ETD) methods, provide highly accurate and stable numerical solutions for stiff evolution equations. However, the cost of these methods is dominated by the computing of matrix exponentials and their products with vectors. The recent development of efficient algorithms for the action of matrix exponentials has increased the effectiveness of ETD methods. To further accelerate the exponential time integration, one can take advantage of parallel and high-performance computing. While direct parallelization at the algebraic level is one possibility, it usually requires a high communication volume since the matrix exponential is global and dense. A different approach is to use domain decomposition techniques and solve a sequence of smaller-sized subdomain problems, in which the matrix exponentials are computed locally and in parallel. In this talk, we present nonoverlapping, noniterative, localized ETD methods for hyperbolic conservation laws and shallow water equations with discontinuous Galerkin (DG) discretization in space. Both mathematical analysis and numerical performance of the proposed methods will be investigated.
Wednesday, July 23, 2025
9:00 - 9:45 AM EDT
Implicit, bound-preserving schemes for degenerate parabolic equations
11th Floor Lecture Hall
Speaker
Christopher Kees, Lousiana State University
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
Two-phase flow in porous media is modeled at large spatial scales by two coupled degenerate, nonlinear parabolic equations, or sometimes a single degenerate parabolic equation. These models arise from averaging microscale free-boundary problems, with both modeling approaches relevant for many natural and industrial processes. We consider specifically water flow in the shallow and variably saturated subsurface environments, such as levees, storm water infiltration systems, and natural coastal environments. Due to the degeneracy of the equations and the density and viscosity ratios for air/water flows, robust, accurate, and efficient discretizations on unstructured meshes have been difficult to extend to practical higher-order discretizations. Recent advances in limiting of higher order continuous finite element methods, alongside advances in analysis of degenerate equations, has opened up routes to genuine advances in unsaturated flow modeling. This work presents work toward higher-order local polynomial approximations for air/water flow, including approaches based on the Richards' equation extending bound-preserving low-order schemes to higher-order methods.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Explicit adaptive solvers for hyperbolic approximations of dispersive water wave models
11th Floor Lecture Hall
Speaker
Carlos Muñoz, King Abdullah University of Science and Technology
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
The shallow water equations (SWEs) are a first-order hyperbolic system used extensively to model free-surface geophysical flows, particularly in hazard assessment contexts. However, their inability to capture dispersive effects limits their accuracy in scenarios involving shorter-wavelength disturbances. Dispersive water wave models, while capable of representing such features, typically lack shock-forming mechanisms that are essential for approximating wave breaking. Additionally, these models often involve high-order and mixed space-time derivatives or differential constraints with no explicit time evolution, posing challenges for efficient large-scale computations. Recent works have proposed hyperbolic approximations to dispersive systems, which offer the potential for efficient, explicit shock-capturing numerical methods. In this work, we develop a hybrid solver that dynamically couples hyperbolic dispersive models with the SWEs across varying flow regimes. Our approach focuses on hyperbolic reformulations of a depth-averaged Euler system and the Serre–Green–Naghdi equations. We evaluate the solver's performance using both laboratory benchmarks and synthetic scenarios.
11:30 AM - 12:15 PM EDT
High Order Semi-implicit schemes for Evolutionary Partial Differential Equations with Higher Order Derivatives
11th Floor Lecture Hall
Speaker
Sebastiano Boscarino, University of Catania
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
The aim of this work is to apply a semi-implicit (SI) strategy to a sequence of one-dimensional time-dependent partial differential equations (PDEs) with high-order spatial derivatives which frequently give rise to stiff problems. We explore the SI approach in the context of implicit-explicit (IMEX) Runge–Kutta (RK), Rosenbrock-type, and IMEX linear multistep (LM) schemes. This strategy provides significant flexibility in handling such equations and enables the construction of simple, linearly implicit schemes without requiring Newton iterations. Furthermore, the SI schemes designed within this framework do not impose severe time step restrictions, which are typically necessary for stability when using explicit methods, i.e., ∆t= O(∆xk) for a k-th order (k ≥ 2) PDE. For spatial discretization, this approach is combined with finite difference schemes, such as WENO schemes. We demonstrate the effectiveness of the proposed methods through various applications to dissipative, dispersive, and biharmonic-type equations. Numerical experiments confirm that the proposed schemes are stable and can achieve optimal orders of accuracy.
12:25 - 12:30 PM EDT
Group Photo (Immediately After Talk)
11th Floor Conference Room
12:30 - 1:45 PM EDT
Lunch/Free Time
1:45 - 2:30 PM EDT
Hyperbolization of Higher-Order PDEs: Theory, Applications, and Numerical Methods
11th Floor Lecture Hall
Speaker
Abhijit Biswas, IIT Kanpur
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
Hyperbolization is a technique that transforms higher-order evolution equations into first-order hyperbolic systems, which become stiff in the relaxation limit. Such constructions have the potential to provide analytical insight and may be useful for efficient computation. In recent years, this kind of hyperbolization has been explored in the context of various higher-order evolutionary PDEs, demonstrating significant potential as a tool for understanding the original equations. In this talk, I will present a systematic framework for uniquely constructing hyperbolized systems for a broad class of linear time-evolution PDEs and discuss how this framework naturally extends to a wide range of nonlinear PDEs of practical interest. I will focus on detailed case studies of two nonlinear PDEs: the Korteweg–de Vries (KdV) equation and the nonlinear Schrödinger (NLS) equation. I will also briefly discuss the traveling wave or solitary wave solutions of the hyperbolized systems and their relationship to those of the original higher-order PDEs. Our findings show that the hyperbolized systems preserve important structural features of the original equations, including the conservation of modified energy for both systems. To address the numerical challenges posed by the resulting stiff systems, we develop fully discrete schemes based on ImEx time stepping methods that are both asymptotic-preserving and energy-preserving. Numerical results support the theoretical findings and highlight the potential of hyperbolization as a framework for better understanding higher-order PDEs.
2:45 - 3:00 PM EDT
Coffee Break
11th Floor Collaborative Space
3:00 - 3:45 PM EDT
Hyperbolic models for dispersive phenomena
11th Floor Lecture Hall
Speaker
Sergey Gavrilyuk, Aix-Marseille Université
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
I present the “augmented Lagrangian” method for dispersive equations admitting variational formulation (the NLS equation, the Benjamin-Bona-Mahoni equation, the Serre-Green-Naghdi equations etc.). This is a kind of penalization method that “transforms” the original equations into an approximate singular system of hyperbolic equations. The advantage of such an approach is obvious: the full range of finite volume methods developed for hyperbolic equations can now be used for dispersive equations. In particular, with the ``hyperbolized model"", the study of the Riemann problem (Gurevich-Pitaevskii's problem) becomes legitimate. The generalization of this method to reversible equations that do not have the variational structure (e.g. the ``conduit"" equation), or to dissipative equations (the Euler equations with thermo-conductivity, or the Cahn-Hilliard equation) is also proposed. I will illustrate this approach with a number of applications: surface wave dynamics, phase transition, etc.
4:00 - 4:45 PM EDT
Space-time parallel solvers based on multigrid-reduction-in-time and operator splitting methods
11th Floor Lecture Hall
Speaker
Laura Portero, Public University of Navarre
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
Many phenomena in science and engineering are governed by evolutionary partial differential equations. In some cases, such as three-dimensional models or long-time simulations, the computational cost becomes a bottleneck for obtaining fast and accurate approximations of the solution. To exploit the capabilities of modern parallel computing architectures, significant research is being conducted in the field of space-time parallel methods. In this talk, we introduce a new class of space-time parallel solvers that combines the iterative multigrid-reduction-in-time (MGRIT) strategy with suitable time-splitting integrators, enabling non-iterative spatial parallelization. We present the main theoretical results and demonstrate the scalability and performance of the proposed approach through a series of numerical experiments.
Thursday, July 24, 2025
9:00 - 9:45 AM EDT
An explicit energy-conserving particle method for the Vlasov-Fokker-Planck equation
11th Floor Lecture Hall
Speaker
Jingwei Hu, University of Washington
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
We present an explicit particle method for the Vlasov-Fokker-Planck equation that conserves energy at the fully discrete level. The method features two key components: a conservative particle discretization for the nonlinear Fokker-Planck operator (also known as the Lenard-Bernstein or Dougherty operator), and an explicit time integrator that ensures energy conservation through an accuracy-justifiable correction. We validate the scheme on several plasma benchmarks, demonstrating its effectiveness. This work is in collaboration with Lee Ricketson and Jiyoung Yoo.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
A Kernel-Based Time-Stepping Framework for Stiff PDEs: Revisiting MOLT with SSP-RK
11th Floor Lecture Hall
Speaker
Sining Gong, Michigan State University
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
Classical approaches for constructing time-stepping methods for stiff partial differential equations (PDEs) often rely on satisfying order conditions or introducing higher-order time derivatives in multi-step or multi-derivative formulations. In this talk, I propose an alternative perspective based on the method of lines transpose (MOLT) framework. By reformulating the spatial derivatives using integral kernel operators, the resulting semi-discrete system can be advanced in time using explicit strong-stability-preserving Runge–Kutta (SSP-RK) methods, while maintaining unconditional linear stability.
This framework enables high-efficiency algorithms with O(N) computational cost and performs especially well under periodic boundary conditions, making it attractive for large-scale simulations of stiff advection-diffusion systems. However, challenges arise when imposing non-periodic boundary conditions, such as Dirichlet or Neumann boundaries, often leading to order reduction. To address this problem, I will present our recent developments that reaches high order for Dirichlet boundary conditions in 1D and 2D, and Neumann conditions in 1D. Numerical examples including stiff advection-diffusion, equations with Dirichlet boundary conditions will be presented to support our results. Future directions will also be presented.
11:30 AM - 12:15 PM EDT
Multirate Integrators for Nonlinearly Partitioned Equations
11th Floor Lecture Hall
Speaker
Tommaso Buvoli, University of California, Merced
Session Chair
Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
Nonlinearly partitioned Runge-Kutta (NPRK) methods are a new family of time integration schemes that generalize additive and component-partitioned Runge-Kutta. Specifically, NPRK methods allow one to treat factors within nonlinearities with differing levels of implicitness. For example, an IMEX-NPRK method can treat a factor of a nonlinear term implicitly while treating the rest explicitly. In this talk, I will give an overview of the NPRK framework and introduce multirate NPRK methods that treat each argument with a different timescale. I will also discuss order conditions and linear stability, and show several numerical experiments that demonstrate the performance of our new multirate methods.
12:30 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
New Features and Applications in the SUNDIALS Library
11th Floor Lecture Hall
Speaker
Carol Woodward, Lawrence Livermore National Laboratory
Session Chair
Jingwei Hu, University of Washington
Abstract
The SUNDIALS library of time integrators and nonlinear solvers has long provided efficient adaptive multistep and multistage time integration methods. New capabilities in SUNDIALS have been added to support multiscale and multiphysics applications, allowing applications the flexibility to switch between single rate and several approaches for handling multiple rate problems. These approaches include several operator splitting as well as multirate methods. This talk will start with a brief overview of the SUNDIALS library and then describe some of the newest features. Results from combustion, climate, and quantum dynamics applications will be presented.
Prepared by LLNL under Contract DE-AC52-07NA27344. LLNL-ABS-2007999.
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:15 PM EDT
Thoughts about time stepping in real-world multiphysics PDE solvers
11th Floor Lecture Hall
Speaker
Wolfgang Bangerth, Colorado State University
Session Chair
Jingwei Hu, University of Washington
Abstract
Many good time stepping methods have been developed over the decades that are well-suited to all sorts of ODEs and DAEs. If one were to implement multiphysics solvers from scratch, one would probably want to consider the use of methods that are well-suited for multirate and DAE problems. In practice, however, code complexity issues and the difference between solving ODEs and spatially-discretized PDEs impose constraints we must consider.
In this talk, I will outline my experience with incorporating modern time steppers into PDE solvers. Specifically, I will consider as an example the ASPECT code (short for ""Advanced Solver for Planetary Evolution, Convection, and Tectonics"") that solves problems relating to the motion of material, energy, and composition in both the Earth crust and the mantle. This is described by a coupled system that involves instantaneous Stokes flow, the advection and diffusion of a temperature field, and the advection and reaction of chemical compositions or other quantities. ASPECT can also simulateously solve for the deformation of the Earth surface, as well as advect particles with the flow field. The resulting time stepping scheme is diverse, but for the most part only uses home-grown time steppers that, for historical reasons, are not well integrated. I will discuss what we do and why, as well as what we do not do and why not. I will also discuss what I think better solutions could be.
4:30 - 5:00 PM EDT
Challenges and open problems
Panel Discussion - 11th Floor Lecture Hall
Session Chair
Jingwei Hu, University of Washington
Panelist
Wolfgang Bangerth, Colorado State University
Panelist
David Ketcheson, King Abdullah University of Science & Technology
Panelist
Chi-Wang Shu, Brown University
Panelist
Carol Woodward, Lawrence Livermore National Laboratory
Abstract
We will discuss the challenges in time-integration and the pressing open problems. This panel will focus on future directions as well as current directions.
Friday, July 25, 2025
9:00 - 9:45 AM EDT
The many faces of stiffness, and how Runge-Kutta methods can overcome their challenges
11th Floor Lecture Hall
Speaker
Benjamin Seibold, Temple University
Session Chair
Zachary J Grant, University of Massachusetts Dartmouth
Abstract
Among the many concepts in numerical analysis for differential equations, "stiffness" appears one of the least straightforward ones to define and characterize. This talk contrasts various forms of stiff limits and associated model problems, and establishes how these different shapes of stiffness manifest in the numerical solution of both ordinary and partial differential equations. Then it is presented which hierarchies of additional order conditions can be formulated that ensure that various classes of Runge-Kutta methods retain their desired convergence rates in the eye of different forms of stiffness.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
Stiff order barriers and Runge-Kutta methods that avoid order reduction with an optimal number of stages
11th Floor Lecture Hall
Speaker
David Shirokoff, New Jersey Institute of Technology
Session Chair
Zachary J Grant, University of Massachusetts Dartmouth
Abstract
This talk presents a solvability theory and construction of Runge-Kutta (RK) methods that avoid order reduction on linear problems via "stiff" order conditions -- referred to as weak stage order (WSO). We first develop general order barriers that bound the maximum allowable stiff order to the number of stages. We then discuss the solvability of the stiff order conditions for schemes with an optimal order and show that for DIRKs and ERKs optimal schemes can be characterized through the solution of a pair of Sylvester equations. Using this theory, we devise schemes that have a positive (linear) SSP coefficient, theoretically optimal accuracy and optimized error constants. We then demonstrate the practical efficacy of the schemes on a suite of test cases. This is joint work with Abhijit Biswas, David Ketcheson, Steven Roberts and Benjamin Seibold.
11:30 AM - 12:15 PM EDT
A Stiff Order Condition Theory for Runge–Kutta Methods: To Semilinearity and Beyond
11th Floor Lecture Hall
Speaker
Steven Roberts, Lawrence Livermore National Laboratory
Session Chair
Zachary J Grant, University of Massachusetts Dartmouth
Abstract
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. This talk will cover recent advancements in addressing order reduction for stiff, semilinear ODEs with an outlook towards broader classes of nonlinear problems. Notably, we do not need to resort to high stage order conditions nor fully implicit methods. We will explore the rich and interesting connections between new stiff order conditions and rooted trees. Numerical experiments confirm we can eliminate order reduction on a wide range of nonlinear ODEs using diagonally implicit Runge–Kutta methods with low stage order.
12:30 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:45 PM EDT
Exploring Stiffness in Tensor-Train Step-and-Truncate Schemes
11th Floor Lecture Hall
Speaker
Rujeko Chinomona, Los Alamos National Laboratory
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
The adoption of tensor networks has led to vast improvements in the tractability of high-dimensional computational simulations by alleviating the curse of dimensionality. Many high-dimensional systems admit data-sparse representations, which can be leveraged to reduce computational storage costs. A popular data-sparse compression technique is the tensor-train (TT) framework. High-dimensional tensors in the TT framework are represented as a sequence (a “train”) of smaller, low-rank tensors, resulting in storage costs that grow only linearly with dimension rather than exponentially. Despite these gains in computational cost when using TT, the same numerical challenges associated with stiffness in full-grid approaches also emerge in low-rank tensor methods. Step-and-truncate algorithms borrow their core time-stepping strategy from full-grid methods while incorporating a low-rank recompression at each step. This preserves manageable storage costs throughout the simulation.
In this talk, I report on a series of numerical experiments in ongoing work involving TT step-and-truncate variants of common stiff integrators. Examples include implicit Runge–Kutta schemes, implicit-explicit (ImEx) Runge–Kutta methods, and multirate techniques. I explore how recompression interacts with the stability and accuracy of each integrator and highlight trade-offs in performance. I conclude by outlining promising directions for integrating existing specialized stiffness solvers with tensor-train tools to develop more efficient methods for high-dimensional stiff systems.
3:00 - 3:30 PM EDT
Coffee Break
11th Floor Collaborative Space
3:30 - 4:15 PM EDT
Explicit-implicit-null (EIN) time-marching for high order PDEs
11th Floor Lecture Hall
Speaker
Chi-Wang Shu, Brown University
Session Chair
David Ketcheson, King Abdullah University of Science & Technology
Abstract
Time discretization is an important issue for time-dependent partial differential equations (PDEs). For the k-th (k is at least 2) order PDEs, the explicit time-marching method may suffer from a severe time step restriction $\tau =O(h^k)$ (where $\tau$ and $h$ are the time step size and spatial mesh size respectively) for stability. The implicit and implicit-explicit (IMEX) time-marching methods can overcome this constraint. However, for the equations with nonlinear high derivative terms, the IMEX methods are not good choices either, since a nonlinear algebraic system must be solved (e.g. by Newton iteration) at each time step. The explicit-implicit-null (EIN) time-marching method is designed to cope with the above mentioned shortcomings. The basic idea of the EIN method is to add and subtract a sufficiently large linear highest derivative term on one side of the considered equation, and then apply the IMEX time-marching method to the equivalent equation. The EIN method so designed does not need any nonlinear iterative solver, and the severe time step restriction for explicit methods can be removed. Coupled with the EIN time-marching method, we will discuss high order finite difference and local discontinuous Galerkin schemes for solving high order dissipative and dispersive equations. For simplified equations with constant coefficients, we perform analysis to guide the choice of the coefficient for the added and subtracted highest order derivative terms in order to guarantee stability for large time steps. Numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy for both one-dimensional and two-dimensional linear and nonlinear equations, including compressible Navier-Stokes equations. This talk is based on joint work with Haijin Wang, Qiang Zhang and Shiping Wang, and with Meiqi Tan and Juan Cheng.
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