How to Preserve Moving Equilibria: Global Flux and Analytical Methods
Talk, Numerical Aspects of Hyperbolic Balance Laws and Related Problems, Ferrara, Italy
Many conservation and balance laws feature families of moving steady states that are essential to preserve at the numerical level. Various techniques have been proposed to achieve this, broadly categorized into well-balanced (WB), and exactly well-balanced (EWB) schemes. WB schemes aim to preserve a numerical equilibrium of the discrete operators, often resulting in super-convergence methods or significantly reduced errors (by several orders of magnitude). EWB schemes, on the other hand, maintain an exact equilibrium (to machine precision) by satisfying an analytical relationship dictated by the equilibrium variables while accounting for the discretization of source terms. In this talk, we compare these methods and analyze the advantages and limitations of each. Specifically, we will consider the well-balanced technique for the Shallow Water equations proposed in [1] and applied in [2] for 1D-like moving equilibria, in contrast to the Global Flux technique used in [3] and [4] for 1D moving equilibria and in [5] for 2D divergence-free moving equilibria.
Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element
Talk, YAMC 2024, Rome, Italy
The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.
Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element
Talk, HONOM 2024, Chania, Crete
The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.
Structure preserving methods via Global Flux quadrature: divergence-free preservation with continuous Finite Element
Talk, Eccomas 2024, Lisbon, Portugal
The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.
Divergence-free Preserving Schemes: what’s wrong in SUPG and how to fix it
Talk, SHARK-FV, Minho, Portugal
The emergence of physical structures and equilibrium solutions, such as divergence-free solutions in contexts like shallow water and magneto-hydrodynamics, poses a significant challenge. A simple linear approximation of such systems that already show these behavior is the linear acoustic system of equations. We focus on Cartesian grid discretizations of such systems in 2 dimensions and in the preservation of stationary solutions that arise due to a truly multidimensional balance of terms, which corresponds to the divergence-free solutions for acoustic systems. Conventional methods, like the continuous Finite Element SUPG, face limitations in maintaining these structures due to the employed stabilization techniques that do not effectively vanish when the discrete divergence is zero. We propose to use the Global Flux procedure, which has proven to be successful in preserving 1-dimensional equilibria GF-WENO, to define some auxiliary variables guiding a suitable discretization of both the divergence and stabilization operators. This approach naturally preserves divergence-free solutions and more intricate equilibria involving various sources. Moreover, this strategy facilitates the identification of discrete equilibria of the scheme that verify boundary or initial conditions. We use the Deferred Correction time discretization, obtaining explicit arbitrarily high order methods. Numerous numerical tests validate the accuracy of our proposed scheme compared to classical approaches. Our method not only (discretely) preserves divergence-free solutions and their perturbations but it also maintains the original order of accuracy on smooth solutions.
IMEX ADER and DeC: arbitrary high order schemes, stability and application to advection–diffusion–dispersion
Talk, INSIDE Indam Workshop, Rome, Italy
Arbitrary Derivative (ADER) [Dumbser et al., JCP, 2008] and Deferred Correction (DeC) [Abgrall, JSC, 2017] are arbitrarily high-order methods developed independently in distinct contexts, yet share notable similarities. Both methods employ an iterative process, incrementing by one the order of accuracy at each step. DeC originated as an ODE solver, then used also for more complicated space-time PDE discretizations, while ADER was initially explored as a PDE solver, particularly in its DG space-time discretization, but it has been investigated also as an ODE solver.
Arbitrary Lagrangian-Eulerian Model Reduction for Advection Dominated Problems and Some Graph Neural Network Ideas
Talk, ENUMATH 2023, Lisbon, Portugal
Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems. Their linear nature does not allow to accelerate the slow decay of the Kolmogorov N-width of these problems. In the recent years, many new nonlinear algorithms and frameworks have been presented to overcome this issue. In this work, we propose a MOR technique for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. The key of the work consists of an arbitrary Lagrangian–Eulerian approach that modifies both the offline and online phases of the MOR process. This allows to calibrate the advected features on the same position and to strongly compress the reduced spaces. We will compare different MOR algorithms between the classical Greedy, EIM and POD and the more recent POD-NN. The calibration map is performed through an optimization process on a training set and then learned through polynomial regression and artificial neural networks for a quick evaluation in the online phase. In the performed simulations we show how the new algorithm defeats the classical method on many equations with nonlinear fluxes and with different boundary conditions. Finally, we compare the results obtained with different calibration maps. We extend the approach also for multiple tracking features and multiple dimensions.
Saving computational costs with efficient iterative ADER methods: p-adaptivity, accuracy results and structure preserving limiters
Talk, Numhyp 2023, Bordeaux, France
Hyperbolic solvers with arbitrarily high order of accuracy are widely used in scientific simulations, but they often come with a high computational cost. In this study, we introduce a modification to the ADER (Arbitrary DERivative) and Deferred Correction (DeC) methods that can save up to half of the computational cost, without sacrificing accuracy. By iteratively increasing the degree of solution reconstruction, our modification provides a natural framework for introducing p-adaptivity in the method, allowing users to adjust the accuracy level according to their goals, cell by cell. Additionally, our approach enables the preservation of solution properties such as positivity, local maximum principle or entropy inequalities, with a very efficient a posteriori limiter. We demonstrate the effectiveness of our method through results applied on ADER-DG and ADER-FV, using the Discrete Optimally increasing Order Method (DOOM) limiter to preserve positivity of density and pressure for compressible Euler and Navier-Stokes equations. Our approach offers a significant computational advantage compared to classical ADER methods, with minimal impact on the accuracy achieved.
Global flux WENO finite volume and other structure preserving schemes for water equations
Talk, SHARK-FV 2023, Minho, Portugal
Water wave equations are known to preserve various equilibria or some specific solutions. The lake at rest steady state is the most well-known equilibrium, and classically many methods are able to preserve such equilibrium. This allows capturing with extreme accuracy the perturbations of such a state.
Model Order Reduction for Advection Dominated (Hyperbolic) Problems in an ALE framework
Talk, PDE Afternoon, Vienna, Austria
Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems. Their linear nature does not allow to accelerate the slow decay of the Kolmogorov N-width of these problems. In the recent years, many new nonlinear algorithms and frameworks have been presented to overcome this issue. In this work, we propose a MOR technique for unsteady parametric advection dominated hyperbolic problems, giving a complete offline and online description and showing the time saving in the online phase. The key of the work consists of an arbitrary Lagrangian–Eulerian approach that modifies both the offline and online phases of the MOR process. This allows to calibrate the advected features on the same position and to strongly compress the reduced spaces. We will compare different MOR algorithms between the classical Greedy, EIM and POD and the more recent POD-NN. The calibration map is performed through an optimization process on a training set and then learned through polynomial regression and artificial neural networks for a quick evaluation in the online phase. In the performed simulations we show how the new algorithm defeats the classical method on many equations with nonlinear fluxes and with different boundary conditions. Finally, we compare the results obtained with different calibration maps.
“Reduced Order Models on a Variational Multi-Scale Model of Navier–Stokes
Talk, CFC, Cannes, France
In this work, we study parametric incompressible flows given by Navier–Stokes equations. At the discrete level, we use continuous finite element (FE) method. The discrete model relies on a variational multi-scale (VMS) approach, which separates the large, sub-filter and small scales. The first two are resolved, while the last is not. The model includes sub-grid eddy viscosity to take care of the interaction between sub-filter and small scales. A local projection stabilization (LPS) term is introduced onto the sub-filter terms to provide stability. This term is based on interpolation and projection operators that penalizes the oscillations on the sub-filter scale. Moreover, in order to deal with no-slip boundary conditions without dramatically refining the mesh close to boundary layers, we resort to wall laws, which take into account the effect of the boundary already at a small distance from the boundary. The computational costs of such simulations, though being faster than other models as the small scales are not resolved, are still large, in particular in the time dependent case. Moreover, dealing with parametric problems in a multi-query context makes the computational burden unbearable. We propose a Galerkin projection of the equations onto a POD-generated reduced basis space as reduced order model (ROM). To take care of the nonlinearities of the problem, different hyper-reduction techniques are studied in order to obtain a reduced model that is independent of the dimension of the FE space. We provide simulations in two dimensions to validate the ROM and to prove the computational advantage of the approach.
A new efficient explicit Deferred Correction framework: analysis and applications to hyperbolic PDEs and adaptivity
Talk, Essentially hyperbolic problems, Ascona, Switzerland
The Deferred Correction is an iterative procedure used to design numerical methods for systems of ODEs, characterized by an increasing accuracy at each iteration. The main advantage of this framework is the automatic way of getting arbitrarily high order methods, which can be put in Runge-Kutta form, based on the definition of subtimenodes in each timestep. The drawback is a larger computational cost with respect to the most used Runge-Kutta methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we remove the unnecessary subtimenodes in all the iterations, introducing interpolation processes between them. We provide the Butcher tableaux of the novel methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in the ODE and PDE settings.
Model order reduction for Friedrichs’ systems: a bridge between elliptic and hyperbolic problems
Talk, MORE 2022, Berlin, Germany
Friedrichs’ systems (FS) K. O. Friedrichs. Comm. Pure & App. Math, 1958 are symmetric positive linear systems of first order PDEs that can describe many well known hyperbolic and elliptic problems in a unified framework. This allows, for example, to pass from one regime to another in different areas of the domain. One of the key ingredients of FS is the possibility of rewriting higher order derivative terms of PDEs through additional variables in the system of equations with only first order terms. This leads to a formulation composed by a linear combination of many block-structured fields $\mathcal{A}^k$ applied to the unknown $z$ and its first order derivatives, i.e., \(\begin{cases} Az=f,\\ (\mathcal{D}-\mathcal{M}) z= 0, \end{cases} \qquad \text{ with } \qquad \begin{cases} Az = A_{(0)}z + A_{(1)}z,\\ A_{(0)}z = \mathcal{A}^0 z,\\ A_{(1)}z = \sum_{k=1}^d \mathcal{A}^k \partial_{x_k} z, \end{cases}\) where $\mathcal{D}$ and $\mathcal{M}$ are boundary fields, one given by the problem and the second used to impose the boundary conditions. Under some coercivity assumptions on the fields, the existence, uniqueness and well-posedness of the problem can be proven in different forms (strong, weak, ultraweak).
Arbitrary High–Order Positivity–Preserving Finite–Volume Shallow–Water scheme without Restrictions on the CFL
Talk, HONOM 2022, Braga, Portugal
This is a talk is about a work in collaboration with Mirco Ciallella, Lorenzo Micalizzi and Philipp Öffner.
Continuous Galerkin high order well-balanced discrete kinetic model for shallow water equations
Talk, Numhyp 2021, Trento, Italy
Kinetic models describe many physical phenomena, inter alia Boltzmann equations, but can also be used to approximate with an artificial relaxation procedure other macroscopic models. We consider the kinetic model proposed by Aregba-Driollet and Natalini, and we modify it in order to approximate shallow water (SW) equations. The difference with the original model stands in the presence of the source term in the SW equations due to the effect of the bathymetry. Thus, the kinetic model by Aregba-Driollet and Natalini must be extended in order to include this term and to maintain the asymptotic convergence to the macroscopic limit of the SW problem.
On modified Patankar schemes and oscillations: towards new stability definitions
Talk, Icosahom 2020, Vienna, Austria
Modified Patankar (MP) schemes are linearly implicit ODE solvers for production destruction problems that guarantee unconditionally the positivity of the solutions and the conservation of the total quantities.
ADER and DeC: Arbitrarily High Order Explicit Methods for hyperbolic PDEs and ODEs
Talk, Oberwolfach, workshop on hyperbolic balance laws, Oberwolfach, Germany
The hyperbolic PDE community has used, in the last years, two powerful techniques that provide arbitrarily high order explicit schemes: ADER (arbitrary derivative) schemes and Deferred Correction (DeC) schemes.
High Order Well-Balanced Discrete Kinetic Model for Shallow Water Equations
Talk, WCCM - Eccomas 2020, Paris, France
In this work, we study a kinetic model that contains stiff relaxation terms in the source. This model can be applied to any non linear hyperbolic problem without source term, that we will call macroscopic problem, to obtain a larger system of equations with linear fluxes and non–linear source terms, the microscopic problem, that converges asymptotically to the original hyperbolic system.
Arbitrary high-order, conservative and positive preserving Patankar-type deferred correction schemes
Talk, Icosahom 2020 MS Online, Online
Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.
Hyperbolic problems: high order methods and model order reduction
Talk, University of Zurich, Zurich, Switzerland
PhD Defense at UZH Zurich. The talk summarize the main topics of the thesis: high order methods for ODEs, hyperbolic PDEs and model order reduction techniques.
Model Reduction for Advection Dominated Hyperbolic Problems in an ALE Framework: Offline and Online Phases
Talk, Analysis Junior Seminars 2019-2020, Trieste, Italy
Model order reduction (MOR) techniques have always struggled in compressing information for advection dominated problems.
ADER and DeC: Arbitrarily High Order Explicit Time Integration Methods
Talk, SAMinar (ETH+UZH), Zurich, Switzerland
When we think about high order time integration, Runge–Kutta (RK) is the most known class of schemes that comes to mind, for historical reasons. Nevertheless, many other techniques have been developed during these years to improve these techniques and to get a generalized form of them.