Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

I will review the existence and uniqueness theory for viscoelasticity of Kelvin-Voigt type with non-convex stored energies. The analysis is based on propagation of $H^1$-regularity for the deformation gradient of weak solutions in two and three dimensions assuming that the stored energy satisfies the Andrews-Ball condition, in particular allowing for non-monotone stresses. By contrast, a counterexample indicates that for non-monotone stress-strain relations (even in 1-d) initial oscillations of the strain lead to solutions with sustained oscllations. In two space dimensions, it turns out that weak solutions with deformation gradient in $H^1$ are in fact unique, providing a striking analogy to corresponding results in the theory of 2D Euler equations with bounded vorticity.

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Hard spheres dynamics is among the finest ones in the scale of models for gas dynamics. Before passing to the limit in order to derive a kinetic model, one needs to understand how many collisions take place, and how strong of weak they are. Collisions can be exponentially many in terms of the number $N$ of particles, but we prove that at most $O(N^2)$ among them can be significant.

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Solutions of the Navier-Stokes equation are not unique. We show that a maximum rate of entropy production is necessary admissibility condition to achieve a physical solution. For incompressible, constant density flow, the resulting solution is probably unique. For variable density flows, multiple distinct flow regimes are possible, and are determined by new physics added at the microscale. Front tracking has a role to enforce some examples of microphysical turbulent mixing regimes. A new multidimensional random choice method appears to be the ideal choice among competing sharp interface algorithms for this purpose.

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Any physical models, while quite different in nature, can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is lack of smoothness as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures including shocks, rarefaction waves and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable high-resolution numerical methods still remains to be an extremely challenging task. In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application, and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest.

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We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by method-of-line Runge-Kutta (RK) methods. The ELDG framework is designed as a generalization of the semi-Lagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general two-dimensional problems. No quadratic-curved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary Lagrangian-Eulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

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We consider a set of scalar conservation laws on a graph. Based on a choice of stationary states of the problem – analogous to the constants in Kruzkhov's entropy condition – we establish the uniqueness and stability of entropy solutions. For two classes of flux functions – either monotone or concave fluxes – we establish the convergence of an easy-to-implement Engquist–Osher-type finite volume method. This is joint work with Markus Musch and Nils Henrik Risebro (University of Oslo).

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Mathematical models introduced to capture the emergent behavior of self-organized systems have brought new challenges in the mathematical community and a lot of attention in the recent years. Most studies on flocking models have been on the behavior of the particle model or the corresponding kinetic formulation and its hydrodynamic formulation that provides in the limit an Euler-type flocking system. This area has been investigated so far in the context of smooth solutions. In this talk, we will discuss a hydrodynamic model of flocking type in the setting of entropy weak solutions. We (i) establish global existence of entropy weak solutions for arbitrary initial data of bounded variation with finite mass confined in a bounded interval and uniformly positive density therein and (ii) show that the entropy solution admits time asymptotic flocking. This is a joint work with Debora Amadori from University of L’Aquila.

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The Onsager conjecture, recently solved by Phil Isett, states that, below a certain threshold regularity, Hoelder continuous solutions of the Euler equations might dissipate the kinetic energy. The original work of Onsager was motivated by the phenomenon of anomalous dissipation and a rigorous mathematical justification of the latter should show that the energy dissipation in the Navier-Stokes equations is, in a suitable statistical sense, independent of the viscosity. In particular it makes much more sense to look for solutions of the Euler equations which, besides dissipating the {\em total} kinetic energy, satisfy as well a suitable form of local energy inequality. Such solutions were first shown to exist by Laszlo Szekelyhidi Jr. and myself. In this talk I will review the methods used so far to approach their existence and the most recent results by Isett and by Hyunju Kwon and myself.

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In this talk we will start with discussion of shock reflection phenomena, and von Neumann conjectures on transition between regular and Mach reflections. Then we describe the results on existence, uniqueness, stability and regularity of global solutions to shock reflection for potential flow, and discuss the techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation in self-similar coordinates, with ellipticity degenerate near a part of the boundary. We will also discuss the open problems in the area.

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Numerical solutions of weak-shock Mach reflections show a sequence of supersonic patches and triple points in a tiny region below the leading triple point. A basic question, with analogs to the existence of shock-free flows over transonic airfoils, is whether it is possible to have only a single shock-free supersonic patch and triple point behind the Mach stem, or must there be multiple triple points. We explore this question using the steady transonic small disturbance equation as the simplest model equation. Assuming the hodograph transformation is invertible near the triple point, we formulate an oblique derivative Tricomi problem for the Tricomi equation as a local description of shock-free flows behind the Mach stem and discuss its solvability.

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The asymptotical stability of wave patterns for viscous conservation laws is an interesting and important problem. Considerable progress on the stability of single shock wave, rarefaction wave in the whole space has been achieved in the past several decades. An interesting question is whether these wave patterns are still stable under periodic spatial perturbation. Compared with the stability problem in the whole space, the solution has no limit at the far fields. In this talk, I will present recent works on the stability of single shock wave and rarefaction wave for viscous conservation laws under periodic perturbations.

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We consider a statistical limit of solutions to the compressible Navier--Stokes system in the high Reynolds number regime in a domain exterior to a rigid body. We investigate to what extent this highly turbulent regime can be modeled by an external stochastic perturbation, as suggested in the related physics literature.To this end, we interpret the statistical limit as a stochastic process on the associated trajectory space. We suppose that the limit process is statistically equivalent to a solution of the stochastic compressible Euler system.Then, necessarily,
1. the stochastic forcing is not active -- the limit is a statistical solution of the deterministic Euler system;
2. the solutions S--converge to the limit;
3. if, in addition, the expected value of the limit process solves the Euler system, then the limit is deterministic and the convergence is strong a.a.
These results strongly indicate that a stochastic forcing may not be a suitable model for turbulent randomness in compressible fluid flows.

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Hyperbolicity-Preserving Well-Balanced Stochastic Galerkin Method for Shallow Water Equations
Dihan Dai, University of Utah
We study the stochastic Galerkin (SG) method for stochastic parameterized shallow water equations. Our work comprises the following aspects: (i) A hyperbolicity-preserving stochastic Galerkin formulation for the shallow water equations using only the conserved variables. (ii) A sufficient condition to preserve the hyperbolicity, which is a stochastic variant of the deterministic positivity condition. (iv) A computationally tractable condition to guarantee the hyperbolicity. (v) A central-upwind scheme that preserves both the hyperbolicity and the well-balanced property at discrete time levels.

A new direct discontinuous Galerkin method with interface correction for compressible Navier-Stokes equations
Mustafa Danis, Iowa State University
We present a new direct discontinuous Galerkin method with interface correction (DDGIC) for solving 2-dimensional compressible Navier-Stokes equations. The new approach is based on modeling the nonlinear diffusion of Navier-Stokes equations as a combination of multiple individual diffusion processes for each equation and conserved variable. This representation of nonlinear diffusion simplifies the numerical flux computation to a great extent since it only requires estimating the numerical flux for gradients of the conserved variables by the simple direct DG numerical flux formula. The new direct DG method is therefore easily extensible to the more general system of equations with nonlinear diffusion. We also demonstrate the high order of accuracy of the new DDGIC method and its ability to capture correct physics through numerical experiments.

Sharp a-Contraction Estimates for Small Extremal Shocks
William Golding, University of Texas at Austin
In the case of systems of conservation laws, it is known that Kruzkov's contraction estimate may fail. However, when comparing an entropy solution to a shock, one may construct a weighted L^2 pseudo-distance, where one may recover the contraction property. In this poster, I will explain some recent work with Alexis Vasseur and Sam Krupa in which we show that the pseudo-distance can be constructed while maintaining precise control over the variation in the weight. This is a key ingredient in the work of Geng Chen, Sam Krupa, and Alexis Vasseur in showing that (for 2 x 2 systems) the conditions of Bressan et al. of Tame Oscillation and Bounded Variations along Space-like Curves are unnecessary for the uniqueness of small BV solutions.

Entropy Methods for Gas Dynamics on Networks
Yannick Holle, RWTH Aachen University
We introduce new coupling conditions for isentropic flow on networks. The new coupling conditions can be derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the generalized Riemann problem holds globally in state space. A numerical example is given in which the new conditions are the only known conditions leading to the physically correct wave types.

Stability of a Nonlocal Conservation Law Modeling Traffic Flow
Kuang Huang, Columbia University
The emerging connected and automated vehicle technologies allow vehicles to perceive and process information in a wide spatial range, which motivates the modeling of traffic flow with nonlocal interactions. For example, conservation laws with nonlocal integral terms were considered in the literature. By conducting stability analysis of one such model, we obtain asymptotic stability of the uniform equilibrium flow under suitable assumptions on how the nonlocal information is utilized. Such results can shed light to the future design of driving algorithms for connected and automated vehicles.

Shallow Water Models with Vertical Velocity Profiles
Julian Koellermeier, KU Leuven
Standard models for shallow water flows like the well-known shallow water equations assume a constant horizontal velocity that does not change in vertical direction. However, this is an unphysical assumption and can lead to wrong results in case of more complex flow fields. In this work, we present a model with vertical velocity profiles, that are expanded in a polynomial series using additional variables. This leads to a so-called moment model that leads to a more realistic description of the flow at the expense of additional equations. Unfortunately, the model equations are not hyperbolic in their original form, which can be mitigated by a hyperbolic regularization that allows to derive real eigenvalues analytically. Numerical tests show the convergence of the model towards reference solutions and a more realistic description of transport properties at the bottom, for example when sediment transport is considered. Further work considers the investigation of equilibrium states, the derivation of well-balancing numerical schemes, and the extension to more advances friction terms by means of a Savage-Hutter model.

On an analytic solution to a 2-D PDE Type of an ideal fluid
Evangelos Nastas, SUNY
This PDE system, whose cardinality of equations exceeds that of its unknowns, is a 2-D equivalent of a general 3-D system, investigating a particular kind of a 2-D PDE of an ideal fluid. It delineates the thermal motion of poly-tropic gas with constant density. Isothermal gas motion with an adiabatic index different to the unit is reduced to the same system. Its study is less complicated in special Lagrangian coordinates. The resulting system consists of linear equations, solved analytically

Adaptive Central-Upwind Scheme on Triangular Grids for the Saint-Venant System
Thuong Nguyen, University of Utah
Joint work with my advisor, Prof. Yekaterina Epshteyn In this work we develop a robust adaptive well-balanced and positivity-preserving central-upwind scheme on unstructured triangular grids for shallow water equations. The numerical method is an extension of the scheme from [Liu et al,J. of Comp. Phys, 374 (2018), pp. 213 - 236]. As a part of the adaptive central-upwind algorithm, we obtain local a posteriori error estimator for the efficient mesh refinement strategy. The accuracy, high-resolution and efficiency of new adaptive central upwind scheme are demonstrated on a number of challenging tests for shallow water models.

Physics-informed Machine Learning of Collective Behaviors
Ming Zhong, Johns Hopkins University
Collective behaviors (clustering, flocking, milling, etc.) are among the most interesting and challenging phenomena to understanding from the mathematical point of view. We offer a non-parametric and physics-based learning approach to discover the governing structure, i.e. the interaction functions between agents, of collective dynamics from observation of the trajectory data. Our learning approach can aid in validating and improving the modeling of collective dynamics.
Having established the convergent properties of our learning approach, we investigate the steady state behavior of the learned dynamics evolved using the estimators inferred from observation data. We then apply our extended learning approach to study the celestial motion of the Solar system using the NASA JPL's modern Ephemeris. We are able to reproduce trajectory data with a precession rate of 558'' per Earth-century for Mercury's orbit. Compared to Newton's theoretical 532'' rate and the observed 575'' rate, we are able to learn portion of the general relativity effect directly from the data. Convergence properties of the extended learning approaches on second-order models are analyzed. Learning collective dynamics on non-Euclidean manifolds has been developed and discussed.

Abstract

Discontinuous Galerkin methods for the wave equation are typically defined by using the fact that the equation can be expressed as a symmetric hyperbolic system. Although these methods can be devised to be high-order accurate, they are naturally dissipative and cannot be used for long-time simulations. We show that it is possible, by taking advantage of the Hamiltonian structure of the wave equation, to overcome this drawback and obtain Discontinuous Galerkin (and other finite element) methods which maintain their original high-order accuracy while conserving the discrete space-discretization energy. We sketch the extension of this approach to other systems of equations with Hamiltonian structure including equations modeling linear and nonlinear elastic, electromagnetic and water wave equations.

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In this talk we present a class of high order unconditionally strong stability preserving (SSP) implicit multi-derivative Runge–Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge–Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order p > 2 is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge–Kutta schemes of order up to p = 4, and IMEX Runge–Kutta schemes of order up to p = 3. For the multi-derivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation. We present numerical results to support the theoretical results, on a variety of problems.

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It was recognized soon after the pioneering works of Riemann and Stokes in the mid-nineteenth century that entropy increases as the gas flows across a shock wave for polyatomic gases. Around 1940's Bethe and Weyl independently formulated the convexity condition for the equivalence of the compressibility of a shock and the entropy increase across it. This was subsequently generalized to the general system of hyperbolic conservation laws by Lax. The situation without convexity is interesting. The Russian school of Oleinik and Krushkov obtained complete results for scalar laws. It is understood now that the existence of entropy for a system is a constitutive hypothesis. Godunov established the relation between the existence of entropy and the symmetric structure of a system. There have been efforts to relate the admissibility conditions for shock waves to the entropy production. For this, we offer a definitive result for shock waves in the Euler equations for compressible media. In this talk, we will survey the historical developments on general systems as well as some exact analysis for the Euler equations.

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In this talk we visit several variations of a standard conservation law. These include the cases such as: (i) the flux function is discontinuous in space and time, (ii) the flux function is nonlocal and contains an integral term, and (iii) combinations of them. We discuss key challenges, some recent results, and applications to traffic flow.

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We show existence of global weak solutions of the Rosensweig model of ferrofluids, using DiPerna-Lions’ theory of compressible fluids. Then, we investigate the relaxation to equilibrium ϵ→0 using the relative entropy method. If the limiting system has a Lipschitz continuous solution, we can show a convergence rate in ϵ, if the limiting system has only a weak solution, we obtain strong convergence of a subsequence in L^2.

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Euler equations under gravitational fields and shallow water equations with a non-flat bottom topography are two prototype hyperbolic conservation laws with source term (also referred as hyperbolic balance laws). They both have various applications in many fields. In this presentation, we will talk about arbitrary order structure preserving discontinuous Galerkin finite element methods which can exactly capture the non-trivial steady state solutions of these models, and at the same time maintain the non-negativity of some physical quantities. Numerical tests are provided to verify the well-balanced property, positivity-preserving property, high-order accuracy, and good resolution for both smooth and discontinuous solutions.

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Conservation laws with nonlocal fluxes have recently drawn considerable attention owing to their applications to several engineering problems, like models of vehicular and pedestrian traffic. They consist of conservation laws where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a (classical) conservation law. In this talk I will overview recent progress on the rigorous justification of this nonlocal-to-local limit. I will mention counter-examples showing that, in general, the solutions of the nonlocal problems do not converge to the entropy admissible solution of the conservation law. On the other hand, the nonlocal-to-local limit have been recently justified, under different assumptions, in the case of anisotropic convolution kernels, which are natural in view of applications to models of vehicular traffic. The talk will be based on joint works with Maria Colombo, Gianluca Crippa and Elio Marconi.

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Weighted essentially non-oscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic conservation laws. They have been applied extensively in computational fluid dynamics and other scientific problems. However, for complicated multidimensional problems, it often leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as fifth order WENO schemes. How to achieve fast simulations by high order WENO methods for solving hyperbolic conservation laws is a challenging and important question. In this talk, I shall present our recent work on applying fast sweeping methods and sparse-grid techniques for efficient computations of WENO schemes. Fast sweeping methods are a class of efficient iterative methods for solving steady state problems of hyperbolic PDEs. They utilize alternating sweeping strategy to cover a family of characteristics in a certain direction simultaneously in each sweeping order. Coupled with the Gauss-Seidel iterations, these methods can achieve a fast convergence speed for computations of steady state solutions of hyperbolic PDEs. We design absolutely convergent fixed-point fast sweeping WENO methods for solving steady state solutions of hyperbolic conservation laws. For high-dimensional problems, sparse-grid techniques are efficient approximation tools to reduce degrees of freedom in the discretizations. We apply the sparse-grid combination technique to fifth order WENO finite difference schemes for solving time dependent hyperbolic PDEs defined on high spatial dimension domains. Extensive numerical experiments shall be shown to demonstrate large savings of computational costs by comparing with simulations using traditional methods for solving hyperbolic conservation laws.

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We address certain challenges related to the analysis of moving boundary problems of parabolic-hyperbolic type. In particular, we focus on a fluid-structure interaction problem between the flow of an incompressible, viscous fluid and a multi-layered poroelastic structure, which behaves as a compressible material. The coupled problem is described by the time dependent Stokes equations, which are coupled to the Biot equations over a poroelastic plate serving as an interface between the free fluid flow and the poroelastic structure. This problem was motivated by the design of a first bioartificial pancreas without the need for immunosuppressant therapy. We will show a recent constructive existence proof for a weak solution to this problem, and a (weak-strong) uniqueness result. This is one of only a handful of well-posedness results in the area of fluid-poroelastic structure interaction problems. The mathematical reasons for this will be discussed, and the impact on the design of a bioartificial pancreas will be shown.

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We will discuss some recent developments of the theory of a contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows. In a first result, in collaboration with Geng Chen and Sam Krupa, we provide some extensions of the Bressan theory for uniqueness of BV solutions in 1D. We show that for 2 × 2 systems, the technical condition, known as bounded variations on space- like curve, is not needed for the uniqueness result. Moreover, we extend the result to a weak/BV stability result (in the spirit of the weak/strong principle of Dafermos) allowing wild perturbations fulfilling only the so-called strong trace property. In a second work in collaboration with Moon-Jin Kang, we consider the stability of 1D viscous shocks for the compressible Navier-Stokes equation, uniformly with respect to the viscosity (JEMS 21'). Thanks to the uniformity with respect to the viscosity, the result can be extended to the Euler equation (the associated inviscid model).This provides a stability result which holds in the class of wild perturbations of inviscid limits of solutions to Navier-Stokes, without any regularity restriction, not even the strong trace property (Inventiones 21'). This shows that the class of inviscid limits of Navier-Stokes equations is better behaved that the class of weak solutions to the inviscid limit problem. Finally, we will present a first multi-D result obtained with Moon-Jin Kang and Yi Wang. We show the stability of contact discontinuities without shear, in the class of inviscid limits of compressible Fourier-Navier-Stokes equation. Note that it is still unknown whether non-uniqueness results can be obtained via convex integration for this special kind of singularity.

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This talk is based on a paper with Matan Mussel ( NIH) that will appear in QAM. The talk provides two new applications of conservation laws in biology. The first is the application of the van der Waals fluid formalism for action potentials. The second is the application of the conservation laws of differential geometry (Gauss–Codazzi equations) to produce non-smooth surfaces representing Endoplasmic Reticulum sheets.

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