Abstract: In the first part of this talk, I will describe a new method for resolving topological singularities, such as coalescence and pinching, in evolving fluid interfaces. The method uses a hybrid sharp–diffuse interface representation, with localized Cahn–Hilliard dynamics during topological events. In the second part, based on joint work with Steve Shkoller, I will describe a numerical method for tracking the formation and subsequent development of one-dimensional shocks, together with the associated weak characteristic discontinuities and cusp structures. This method uses a moving mesh adapted to the evolving characteristic wave families. In both settings, I will demonstrate orders-of-magnitude gains in accuracy and efficiency over conventional numerical schemes.
]]>Abstract: This work addresses all-speed flows, and in particular low-Mach-number flows, for the numerical approximation of the Kapila et al. [2] multiphase flow model. This model is valid for fluid mixtures evolving in mechanical equilibrium but out of thermal equilibrium and is efficient for computing material interfaces separating miscible and non-miscible fluids. In this context, the interface is treated as a numerically diffused zone that captures all present waves (shocks, expansion waves). The same flow model can be used to solve cavitating and boiling flows [3]. Many applications involving liquid–gas interfaces and cavitating flows span a wide range of Mach numbers, from 10E−3 to supersonic (and even hypersonic) conditions with respect to the mixture sound speed. Therefore, it is important to develop numerical methods free of Mach-number restrictions. To do this, we build a preconditioned Riemann solver and embed it in the Godunov explicit scheme. This method converges to exact solutions but requires very small time steps to be efficient. We then derive an implicit version, first in one dimension and then for 2D unstructured meshes. We then address two-phase flow preconditioning in the framework of the Saurel et al. [4] algorithm. We detail the necessary modifications to the preconditioned Riemann solver. We demonstrate convergence of both single-phase and two-phase numerical solutions using single-phase and two-phase steady nozzle flow solutions. Finally, the method is illustrated by computing real cavitating flows in Venturi nozzles. The model and method reproduce vapor pocket size and instability frequencies without using any adjustable parameter.
Abstract: We discuss the global existence of weak (entropy) solutions for two distinct classes of hyperbolic systems by constructing novel viscous regularizations. In the first part of the talk,, we focus on a class of 2 × 2 Keyfitz-Kranzer systems arising in thin film flows. Because the standard diagonal viscous approximation fails to yield the appropriate mathematical structure, we instead construct a novel viscous regularization inspired by lubrication equations. This approach provides the critical compactness estimates necessary to establish the existence of weak entropy solutions via the vanishing diffusion limit [1].
In the second part of the talk, we generalize these results to n×n systems of a broader KeyfitzKranzer class covering several important physical models. We introduce a new artificial viscous regularization aligned with the nonlinear field, which yields a parabolic equation for the Riemann invariant associated with the nonlinear field, coupled with transport equations for the remaining Riemann invariants. This structure drives the compactness argument and yields global existence of weak solutions for the inviscid system [2].
(Based on the joint works with Christian Rohde (University of Stuttgart), Phillipp Öffner (TU Clausthal) and Lilu Sahu(IIT Kharagpur))
Abstract: Discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws produce semi-discrete systems M dU/dt = L(U) whose time evolution is dominated by repeated evaluation of the spatial operator L(). The standard DG approach assembles dense local element matrices cell-by-cell through matrix-matrix products. Matrix-free evaluation replaces these with matrix-vector operations via sum factorization. Because the basis is a product of one-dimensional polynomials the entire operator is written as a sequence of tensor-product contractions.
Per degree of freedom, the work of matrix-free operator evaluation grows as O(k) in the polynomial degree k, versus O(k^d) for matrix-based assembly. A further ingredient is SIMD vectorization: deal.II batches cells into SIMD lanes, so the same flux function evaluates four or more cells simultaneously.
]]>Abstract: We present two optimisations in the context of high order continuous Galerkin methods on curvilinear meshes. First, we exploit the Kronecker delta structure of Lagrange polynomials at Gauss Lobatto Legendre (GLL) quadrature points to eliminate vanishing terms in gradient penalty computation, reducing assembly cost by a factor of four in three dimensions with degree 4. Second, we implement Kopriva’s invariant curl form to enforce the discrete metric identity for curvilinear coordinate mappings, achieving errors at machine precision across smooth and randomly perturbed grids, in contrast to the direct method which incurs large errors for mapping degrees exceeding two.
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Particulate processes such as crystallization, granulation, aerosol dynamics, and polymerization, among others are governed by particle populations whose size distributions evolve through nucleation, growth, aggregation, and breakage mechanisms. Tracking individual particles is infeasible at industrially relevant scales, so these systems are modelled in the continuum framework using population balance equations (PBEs), a class of integro-differential equations that describe the evolution of the particle size distribution.
While PBEs offer a tractable and mechanistically grounded description, solving them numerically is far from straightforward. The nonlocal integral terms associated with aggregation and breakage introduce strong coupling across the particle size domain, making the design of accurate and stable numerical methods difficult. Moreover, standard discretization techniques, including widely used finite volume schemes, often fail to preserve important moments of the distribution. This limitation is significant because such moments are directly related to physically meaningful and industrially relevant quantities.
This talk will focus on the development of weighted finite volume schemes for PBEs that are specifically designed to preserve mass together with selected higher-order moments, while maintaining computational efficiency and numerical stability. The underlying mathematical formulation and construction of these moment-preserving schemes will be discussed in detail, with emphasis on consistency, conservation properties, and implementation aspects. Numerical results will be presented for benchmark problems possessing analytical solutions, as well as for application- oriented test cases arising in particulate process engineering, demonstrating the accuracy, robustness and effectiveness of the proposed methods.
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In this talk, we are primarily interested in the study of different aspects of solutions to some system of partial differential equations known as system of hyperbolic conservation laws. Conservation laws typically assert that the rate of change of conserved quantity in a region is governed by a flux function. Many mathematical models such as Euler equations of gas dynamics, shallow water equations, combustion equations, nonlinear elasticity, traffic flow equations, chromatography equations, magnetohydrodynamics, etc can be described by a system of conservation laws. Existence of nonclassical solution (using flux perturbation method, self-similar vanishing viscosity method), limiting behaviour (zero relaxation limit) as well as large time behaviour of solution to Cauchy problem for some system of partial differential equations are explored.
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Accurate numerical simulation of multiscale fluid systems is a central challenge in computational mathematics and physics. In particular, the barotropic Euler system exhibits multiscale behavior in the low Mach number regime, which can render standard methods inefficient or unstable due to stiffness. This motivates the development of numerical schemes that are both efficient and robust across multiscale regimes.
In this talk, I present implicit–explicit (IMEX) numerical methods for the barotropic Euler equations that are asymptotic-preserving and fully discrete energy-stable, ensuring both consistency with the incompressible limit and nonlinear stability. I discuss the construction of these methods, provide consistency analysis through discrete energy stability, and present rigorous error estimates that remain uniform with respect to the Mach number. To demonstrate their practical effectiveness, I will also show numerical experiments illustrating energy stability, accuracy, and convergence in challenging low Mach number regimes. While this work focuses on the barotropic Euler system, it provides insights that may guide the development of robust, structure-preserving methods for related multiscale fluid systems.
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This work examines numerical methods applied to the Boussinesq system of equations, which is widely used in real–world applications to analyze fluid flow and heat transfer phenomena. The system consists of the Navier–Stokes system of equations coupled with the convection–diffusion equation for temperature. To begin with, a three–step two–grid continuous Galerkin (CG) finite element method (FEM) is proposed, combining Newton’s iteration for spatial discretization with the backward Euler (BE) method for temporal discretization.
The two–grid technique is an efficient approach for solving nonlinear problems, where the solution is first computed on a coarse mesh and then used to linearize the problem on a finer mesh. This method helps achieve accurate approximations while mitigating the high computational cost associated with the nonlinear nature of the model. Subsequently, a discontinuous Galerkin (DG) method is investigated, employing DG method for spatial discretization and BE method for time discretization. This is followed by a two–grid DG FEM approach with BE method for time discretization. The DG method is a powerful technique for solving partial differential equations, offering high accuracy, local conservation, and stability even on irregular meshes. It excels at capturing complex phenomena and supports varying polynomial degrees across elements.
Additionally, a local discontinuous Galerkin (LDG) method for spatial discretization combined with BE method for time discretization is explored. As a variant of the DG method, the LDG method improves stability and conservation by applying local penalty terms at element boundaries, ensuring continuity and local conservation. For all proposed numerical schemes, stability and optimal error estimates are derived for velocity, pressure, and temperature. Finally, numerical experiments are conducted to evaluate the performance of the proposed methods.
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Many classical discretization techniques for partial differential equations rely on explicitly constructed finite-dimensional approximation spaces defined on simplicial or tensor-product meshes. While highly successful, this paradigm becomes restrictive when one seeks robust and stable methods on general polygonal or polyhedral meshes arising from adaptivity, mesh agglomeration, or complex geometries.
The Virtual Element Method (VEM) offers a principled framework for the discretization of partial differential equations on arbitrary meshes without re- quiring explicit knowledge of basis functions inside mesh elements. Instead, the method is formulated through carefully chosen degrees of freedom, computable projection operators, and discrete bilinear forms designed to satisfy polynomial consistency and stability properties.
This talk presents a conceptual introduction to the Virtual Element Method for elliptic problems, emphasizing the underlying approximation and stability mechanisms rather than implementation details. Starting from a model problem, we explain how consistency is enforced through polynomial projections and how stabilization controls the non-polynomial components of the virtual space. We also discuss convergence properties, mesh-regularity considerations, and the flexibility of the framework in comparison with classical finite element approaches. The talk concludes with a brief overview of applications and current research directions in virtual element technology.
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