Recent Advances in Numerical PDEs

A workshop, held at the University of Pittsburgh,
May 5-6, 2025

Due to the prevalence of PDEs in engineering, physics, biology, medicine and the social sciences, numerical solutions have become an indispensable tool in scientific exploration. Their significance lies not only in their descriptive capacity, but also their predictive power to help us understand aspects of the physical world and its complex systems.

The workshop aims to promote the direct interaction between experts from different communities, link rigorous numerical analysis and analysis of PDEs with current problems of impact.
Specific topics may include:
- Structure preserving discretizations, emphasizing the enforcement of mass conservation at the discrete level;
- Penalty and artificial compression methods;
- Partitioned and monolithic time-stepping, Structure-fluid interaction;
- Inverse Problems, Data-Driven Modeling & Scientific Computation.

There is no charge for attending the workshop.

Tentative Participant List

• Isabel Barrio Sanchez (University of Pittsburgh)
• Jeffrey Borggaard (Virginia Tech)
• Martina Bukač (Notre Dame)
• Yanzhao Cao (Auburn University)
• Victor DeCaria (Naval Nuclear Laboratory)
• Vince Ervin (Clemson University)
• Qiwei Feng (University of Pittsburgh)
• Max Gunzburger (Florida State University)
• Lili Ju (University of South Carolina)
• Monica Morales Hernandez (Adelphi University)
• Jiabao Nie (Notre Dame)
• Wenlong Pei (Ohio State)
• Janet Peterson (Florida State University)
• Leo Rebholz (Clemson University)
• Maicon Ribeiro Correa (University of Campinas)
• Michael Schneier (Naval Nuclear Laboratory)
• Giselle Sosa Jones (Oakland University)
• Miroslav Stoyanov (Oak Ridge National Laboratory)
• Hoang Tran (Oak Ridge National Laboratory)
• Alessandro Veneziani (Emory University)
• Hans-Werner van Wyk (Auburn University)
• Noel Walkington (Carnegie Mellon)
• Guannan Zhang (Oak Ridge National Laboratory)
• Yanzhi Zhang (Missouri University of Science and Technology)
• Lizette Zietsman (Virginia Tech)

Organizers

John Burkardt, William Layton, Catalin Trenchea

Location

The conference will be held in the Cathedral of Learning, room 332.

Schedule
Monday, May 5
9:30am	Welcome Coffee
10:00am	Alessandro Veneziani Data Assimilation, Optimization and Reduced Order Modeling in Cardiovascular Mathematics
10:50-11:10am	Coffee break
11:10am	Leo Rebholz Accelerating solvers for fluids with (continuous) data assimilation
11:40am	Guannan Zhang Dynamic Generative AI for Nonlinear Data Assimilation
12:10-2:10pm	Lunch break
2:10pm	Jeffrey Borggaard Nonlinear Feedback Control of Polynomial Systems
2:40pm	Lili Ju Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations
2:50-3:10pm	Coffee break
3:10pm	Yanzhi Zhang Numerical Studies of Anomalous Diffusion in Heterogeneous Media
3:40pm	Miroslav Stoyanov Sparse Tensor Kronecker Operations for High-Dimensional Discontinuous Galerkin
4:10-4:30pm	Coffee break
4:30pm	Maicon Ribeiro Correa Study of Sequential Coupling Strategies for a Black-Oil Model in Poroelastic Media
5:00pm	Giselle Sosa Jones An energy-stable time stepping method for two-phase flow problems in porous media

Tuesday, May 6
9:00am	Hoang Anh Tran Surrogate modeling for MHD flows in liquid metal fusion blankets
9:30am	Jiabao Nie Linear Stability Analysis and Numerical Investigation of Bioconvection
10:00-10:10am	Coffee break
10:10am	Isabel Barrio Sanchez Second-order in time decoupled time stepping methods for heat transfer
10:40am	Qiwei Feng Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect
11:10-11:20pm	Coffee break
11:20	Wenlong Pei Partition conservative, variable step, second-order method for magnetohydrodynamics in Elsässer variables

Map

A map of various landmarks on campus can be found here.

Speaker Information

• Isabel Barrio Sanchez (University of Pittsburgh)
  Second-order in time decoupled time stepping methods for heat transfer
  Many physical systems involve heat transfer between two interacting subdomains, particularly when the domains have different thermal diffusion coefficients. Such scenarios occur in applications like ocean-atmosphere coupling, turbine blade cooling, road surface defrosting, and radiative heat exchange. In these cases, heat behavior at the interface can differ significantly. We propose and analyze a second-order accurate time-stepping method with subiterations to model these coupled systems. We prove the convergence and stability of the method in the linear case and validate its second-order accuracy through numerical simulations.
• Jeffrey Borggaard (Virginia Tech)
  Nonlinear Feedback Control of Polynomial Systems
  We develop and present nonlinear feedback control laws for polynomial systems by computing polynomial approximations to Hamilton-Jacobi-Bellman equations. We emphasize polynomial models that arise in discretizations of control problems involving the Navier-Stokes equations but also consider higher-degree and bilinear (state and control variables) terms. To demonstrate their effectiveness, we implement a quadratic feedback control for the fluidic pinball problem, a benchmark problem that seeks to eliminate the vortex shedding behind three cylinders using cylinder rotation as the actuation mechanism. Numerical simulations of this feedback law (a closed-loop simulation performed using FEniCS) demonstrate that we can completely stabilize the steady-state solution (i.e. no vortex shedding) over a range of low Reynolds number flows where linear feedback laws fail. Thus, quadratic feedback expands the domain of attraction for this closed-loop system. This is joint work with Hamza Adjerid and Ali Bouland.
• Qiwei Feng (University of Pittsburgh)
  Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect
  Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High-order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high-order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method is sixth-order consistent for a constant wavenumber, and fifth-order consistent for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $\kappa h$ is near $1$ with $\kappa$ being the wavenumber and $h$ the mesh size. This is joint work with Bin Han, and Michelle Michelle.
• Lili Ju (University of South Carolina)
  Dynamically regularized Lagrange multiplier schemes with energy dissipation for the incompressible Navier-Stokes equations
  In this talk, we present efficient numerical schemes based on the Lagrange multiplier approach for the Navier-Stokes equations. By introducing a dynamic equation (involving the kinetic energy, the Lagrange multiplier, and a regularization parameter), we form a new system which incorporates the energy evolution process but is still equivalent to the original equations. Such nonlinear system is then discretized in time based on the backward differentiation formulas, resulting in a dynamically regularized Lagrange multiplier (DRLM) method. First- and second-order DRLM schemes are derived and shown to be unconditionally energy stable with respect to the original variables. The proposed schemes require only the solutions of two linear Stokes systems and a quadratic equation at each time step. Moreover, with the introduction of the regularization parameter, the Lagrange multiplier can be uniquely determined from the quadratic equation, even with large time step sizes, without affecting accuracy and stability of the numerical solutions. Fully-discrete energy stability is also proved with the Marker-and-Cell (MAC) discretization in space. Various numerical experiments in two and three dimensions verify convergence and energy dissipation as well as demonstrate accuracy and robustness of the proposed DRLM schemes.
• Jiabao Nie (University of Notre Dame)
  Linear Stability Analysis and Numerical Investigation of Bioconvection
  Bioconvection refers to the spontaneous formation of large-scale convection patterns in suspensions of swimming microorganisms, such as algae and bacteria. Mathematical modeling serves as a crucial tool for describing, analyzing, and predicting the complex interactions between microorganisms and the surrounding fluid. A fundamental question in applications is to determine the critical conditions under which bioconvection initiates or remains stable. In this talk, I will present a linear stability analysis conducted to identify these critical conditions. A major challenge arises from the presence of non-constant coefficients in the linearized system. To address this, various approximations are introduced to facilitate analytical treatment. Additionally, computational methods are employed to solve the original linearized system without approximation. The analytical and numerical results are compared, and their consistency is further validated against full nonlinear numerical simulations.
• Wenlong Pei (Ohio State University)
  Partitioned conservative, variable step, second-order method for MHD in Elsasser variables
  Magnetohydrodynamics (MHD) describes the interaction between electrically con- ducting fluids and electromagnetic fields. We propose and analyze a symplectic, second-order algo- rithm for the evolutionary MHD system in Elsässer variables. We reduce the computational cost of the iterative non-linear solver, at each time step, by partitioning the coupled system into two sub- problems of half size, solved in parallel. We prove that the iterations converge linearly, under a time step restriction similar to the one required in the full space-time error analysis. The variable step algorithm unconditionally conserves the energy, cross-helicity and magnetic helicity, and numerical solutions are second-order accurate in the L2 and H1-norms. The time adaptive mechanism, based on a local truncation error criterion, helps the variable step algorithm balance accuracy and time efficiency. Several numerical tests support the theoretical findings and verify the advantage of time adaptivity.
• Leo Rebholz (Clemson University)
  Accelerating solvers for fluids with (continuous) data assimilation
  We consider nonlinear solvers for the incompressible, steady Navier-Stokes equations in the setting where partial solution data is available, e.g. from physical measurements or sampled solution data from a (too big to send) very high-resolution computation. The measurement data is incorporated/assimilated into the solver through a nudging term addition that penalizes at each iteration the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented at each time step for time dependent dissipative PDEs. For a Picard solver, we quantify the acceleration provided by the data in terms of the density of the measurement locations and the level of noise in the data. For Newton, we show how the convergence basin for the initial condition is expanded as more data is assimilated.
  Numerical tests illustrate the results. While the setting is for Navier-Stokes, the ideas are applicable to solvers for a wide range of nonlinear systems.
• Maicon Ribeiro Correa (University of Campinas)
  Study of Sequential Coupling Strategies for a Black-Oil Model in Poroelastic Media
  In this talk, we briefly discuss a Black-Oil model in Poroelastic media, where the conservation of the corresponding component densities replaces the traditional black-oil conservation volume equations at standard conditions. The extended flow equations, describing the movement of the aqueous, oleic, and gaseous phases, incorporate new complex features associated with transient porosity and are rewritten in a proper Lagrangian formulation. Such a model overcomes the necessity of keeping track of the dynamic hysteretic behavior of the bubble point pressure, treating the phase appearance by assessing the excess concentration relative to the saturation limit in the oil phase. Then, we study two sequentially coupled fixed-stress algorithms, where the subsystems for flow, transport and deformation are solved iteratively by appropriate numerical schemes. The potential of the proposed coupling strategies is illustrated in numerical simulations of black-oil flow problems in poroelastic media.
• Giselle Sosa Jones (Oakland University)
  An energy-stable time stepping method for two-phase flow problems in porous media
  Modeling the flow of liquid, aqueous, and vapor phases through porous media is a complex and challenging task that requires solving nonlinear coupled partial differential equations. In this talk, we propose a second-order accurate and energy-stable time discretization method for the two-phase flow problem in porous media. We prove the convergence of the subiterations to resolve the nonlinearity, and show that the time-stepping method mimics the energy balance relation that the continuous problem satisfies. Our spatial discretization uses an interior penalty discontinuous Galerkin method, for which we establish the well-posedness of the discrete problem and provide error estimates under certain conditions on the data. We validate our method through numerical simulations, which show that our approach achieves the expected theoretical convergence rates. Furthermore, the numerical examples highlight the advantages of our time discretization over other time discretizations.
• Miroslav Stoyanov (Oak Ridge National Laboratory)
  Sparse Tensor Kronecker Operations for High-Dimensional Discontinuous Galerkin
  Sparse grids are a family sampling and discretization techniques for multidimensional function approximations and partial differential equations. The grids are constructed from hierarchical superposition of multidimensional tensors of single dimensional basis functions and many operations resemble linear algebra using classical Kronecker matrices. Kronecker theory is a powerful tool for tensor operations that allows us to combine like-terms and reduce the overall computational cost by a factor exponential in the number of dimensions. However, the same like-term relationship is lost in a general sparse context. We present several sufficient conditions that allow us to recover the Kronecker cost-reduction in a hierarchical sparse grid context. We apply our strategy to Discontinuous Galerkin discretization of a high-dimensional PDEs and show over 100x speedup for even moderate dimensions.
• Hoang Anh Tran (Oak Ridge National Laboratory)
  Surrogate modeling for MHD flows in liquid metal fusion blankets
  Liquid metal blankets plays a central role in future fusion reactors. Designing a new blanket or improving characteristics of a current blanket concept requires magnetohydrodynamic (MHD) codes to construct a suitable blanket solution. However, this often comes with an extremely high computational cost. Surrogate modeling is a modern approach which promises to mitigate this challenge by developing reliable, high-quality surrogates for MHD systems. Once trained, the surrogate models can be deployed to replace high-fidelity simulations, where the users can inquire directly to extract the system statistics and quantities of interest, virtually at no cost. In this work, we present an investigation and initial evaluation of three popular surrogate modeling methods for MHD flows in liquid metal fusion blankets, namely, sparse grid interpolation, sparse polynomial expansion and Gaussian process. These methods, which use interpolation or regression fits of high-fidelity simulations, are well-suited for the blanket problem, as they generally perform well with limited training data -- a likely scenario given the cost of high-fidelity MHD simulations. We demonstrate the performance of surrogate modeling on a simple case of fully-developed MHD flow in a rectangular duct, where the analytical solutions exist, and then a more complex test case of MHD flow in a duct with expansion, where the Hartmann and Reynolds numbers reach up to 10000. The insights and experience gained from this study are expected to serve as an important foundation for developing the approach to more practical blanket applications.
• Alessandro Veneziani (Emory University)
  Data Assimilation, Optimization and Reduced Order Modeling in Cardiovascular Mathematics
  Reduced-order models (ROM) provide practical solutions to problems that were once considered too computationally expensive. In Cardiovascular Mathematics, surgical optimization takes Personalized Medicine to an unprecedented level. Meanwhile, Data Assimilation (DA) may play a pivotal role in bridging theory with clinical practice. DA involves a set of techniques that integrate mathematical models with measurements to improve our understanding of specific problems. In clinical settings, efficient methods that combine physics-informed models with available data-i.e., background and foreground knowledge-can offer a deeper and more precise understanding of a patient's condition. Although the mathematical formulations of these problems are well-established, high computational costs have historically prevented their translation into clinical practice. In fact, these problems require the timely solution of constrained optimization, fluid and/or structural mechanics; this is not feasible without methods that reduce computational costs. ROM presents a viable workaround, with the potential for significant impact in medicine. In this talk, we will explore applications in pediatric surgery (specifically Total Cavopulmonary Connection), as well as coronary diseases (including stent optimization and wall shear stress data assimilation). We will also discuss the use of Proper Orthogonal Decomposition and the potential role of Physics-Informed Neural Networks in these contexts.
• Guannan Zhang (Oak Ridge National Laboratory)
  Dynamic Generative AI for Nonlinear Data Assimilation
  We propose an ensemble score filter (EnSF) for solving high-dimensional nonlinear filtering problems with superior accuracy. A major drawback of existing filtering methods, e.g., particle filters or ensemble Kalman filters, is the low accuracy in handling high-dimensional and highly nonlinear problems. EnSF addresses this challenge by exploiting the score-based diffusion model, defined in a pseudo-temporal domain, to characterize the evolution of the filtering density. EnSF stores the information of the recursively updated filtering density function in the score function, instead of storing the information in a set of finite Monte Carlo samples (used in particle filters and ensemble Kalman filters). Unlike existing diffusion models that train neural networks to approximate the score function, we develop a training-free score estimation method that uses a mini-batch-based Monte Carlo estimator to directly approximate the score function at any pseudo-spatial–temporal location, which provides sufficient accuracy in solving high-dimensional nonlinear problems while also saving a tremendous amount of time spent on training neural networks. High-dimensional Lorenz-96 systems are used to demonstrate the performance of our method. EnSF provides superior performance, compared with the state-of-the-art Local Ensemble Transform Kalman Filter, in reliably and efficiently tracking extremely high-dimensional Lorenz systems (up to 1,000,000 dimensions) with highly nonlinear observation processes.
  Even though the proposed framework is tested only on a benchmark surface quasi-geostrophic (SQG) turbulence system, it has the potential to be combined with existing AI-based foundation models, making it suitable for future operational implementations.
• Yanzhi Zhang (Missouri University of Science and Technology)
  Numerical Studies of Anomalous Diffusion in Heterogeneous Media
  The transition between different diffusion states is a common feature in highly heterogeneous, fractal-like media. This phenomenon is observed across various fields, including dusty plasma, living human cells, hydrology, proteins, and polymer liquids. The application of nonlocal fractional models to study anomalous diffusion transitions is a relatively recent advancement. In particular, the variable-order fractional Laplacian plays a key role in understanding heterogeneous systems. In this talk, we will explore the challenges and current progress in studying anomalous diffusion in heterogeneous media. We will also introduce several numerical methods for solving the variable-order Laplacian and discuss their properties. Additionally, we will apply these methods to investigate the solution behaviors of variable-order fractional PDEs in different contexts. These findings will provide valuable insights into the understanding and application of anomalous diffusion in heterogeneous media.

Directions

The Pittsburgh International Airport (PIT) is 20.5 miles away from the University of Pittsburgh (25-45 minutes driving time depending on the time of day). There are several options to get to campus from the airport.

• Express Shuttle Service (1-800-991-9890). A shuttle service between the airport and the downtown Pittsburgh and Oakland areas. The fare is $25 one way and $43 round trip.
• SuperShuttle (1-800-258-3826). Another shuttle service that serves the airport and the Oakland/Downtown area.
• Public Bus Transportation with Port Authority Transit 28X Airport Flyer. The trip cost is $3.75 one way and exact change is required. The bus runs every 30 minutes from the airport at gate #6, and the duration of the trip is typically 50 minutes. See the flyer for more details.
  

See this link for a self-guided virtual tour of the University of Pittsburgh campus.
The Transit app, available for smart phones, is useful for moving around the city using the bus system, with an up-to-date schedule of the buses, also allowing to buy tickets online.

Parking

If you need to park on campus, you might want to park at the following garage:

• Soldiers and Sailors (SO): The rates are 0-1 hour $4; 1-2 hours $5; 2-4 hours $7; 4-6 hours $8; 6-8 hours $10; 8-10 hours $12; Over 10 hours $14.
• All garages are listed on this map.

Dining Options

There are several dining options throughout the Oakland area. I personally recommend Ali Baba (Middle Eastern), Peter's Pub (American), Fuel and Fuddle (American), or the Spice Island Tea House (Asian). The locations of these restaurants are given in this map.

Contact:

For questions please contact Catalin Trenchea.

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Acknowledgement:

The organizers gratefully acknowledge the help and financial support provided by The University of Pittsburgh Mathematical Research Center