arXiv subjects

Alina Chertock

Publications and source records attributed to Alina Chertock.

At least 19 recordsLinked to original sources

arXiv2026

A New Asymptotic-Preserving Dual Formulation Finite-Volume Method for the Compressible Euler Equations

The paper focuses on the development of numerical methods for the compressible Euler equations. It is well-known that if the Mach number is small, the system becomes stiff and hence explicit schemes suffer from severe time-step restrictions, making them inefficient or even impractical. Our objective is to develop an asymptotic preserving (AP) scheme that remains uniformly accurate and stable across all Mach numbers. Instead of the conservative hyperbolic flux splitting approach, which is widely used to design AP schemes, we consider a primitive (nonconservative) formulation and introduce a nonconservative hyperbolic splitting. The resulting system is discretized using a semi-implicit approach: the stiff part is handled semi-implicitly using second-order central differences, while the nonstiff part is treated explicitly using a second-order path-conservative central-upwind discretization. A key feature of our method is that the pressure at each time level is computed by solving a well-posed Poisson-type elliptic equation, thereby enforcing the AP property. Simultaneously, we evolve the conservative form of the system using a semi-discrete central-upwind (CU) scheme. At the end of each stage of the time discretization, we perform a special post-processing that selects the appropriate numerical solution depending on the Mach number. This guarantees that in low-Mach-number regimes, the solution is obtained by the AP nonconservative scheme, while in higher-Mach-number regimes, a sharp and physically relevant solution is computed by the conservative CU scheme. Numerical experiments confirm that the proposed AP scheme achieves the expected second order of accuracy and that the time-step constraint is independent of the Mach number, making it a robust and efficient alternative to conventional explicit methods.

math.NA↗

arXiv2026

An Asymptotic-Preserving Dual Formulation Finite-Volume Method for the Thermal Rotating Shallow Water Equations

We propose a new second-order asymptotic-preserving (AP) dual formulation finite-volume (DF-FV) method for the thermal rotating shallow water (TRSW) equations. The TRSW system models geophysical flows characterized by horizontal temperature/density variations, exhibiting multi-scale dynamics due to the coexistence of fast rotational waves and slower advective processes. To efficiently address challenges associated with the multiscale nature of the TRSW system, we follow the DF-FV framework and develop a DF-FV method, in which both the conservative and nonconservative (primitive) forms of the equations are simultaneously solved, allowing the method to exploit the complementary strengths of each representation across different flow regimes. The primitive formulation is better suited for preserving the correct asymptotic behavior in nearly thermal quasi-geostrophic (TQG) regimes characterized by a low Rossby number, while the conservative formulation is essential for robust shock capturing in high-Rossby-number regimes, in which nonconservative discretizations may fail to converge to physically relevant weak solutions.

math.NA↗

arXiv2026

New Adaptive Numerical Methods Based on Dual Formulation of Hyperbolic Conservation Laws

In this paper, we propose an adaptive high-order method for hyperbolic systems of conservation laws. The proposed method is based on a dual formulation approach: Two numerical solutions, corresponding to conservative and nonconservative formulations of the same system, are evolved simultaneously. Since nonconservative schemes are known to produce nonphysical weak solutions near discontinuities, we exploit the difference between these two solutions to construct a smoothness indicator (SI). In smooth regions, the difference between the conservative and nonconservative solutions is of the same order as the truncation error of the underlying discretization, whereas in nonsmooth regions, it is ${\cal O}(1)$. We apply this idea to the Euler equations of gas dynamics and define the SI using differences in the momentum and pressure variables. This choice allows us to further distinguish neighborhoods of contact discontinuities from other nonsmooth parts of the computed solution. The resulting classification is used to adaptively select numerical discretizations. In the vicinities of contact discontinuities, we employ the low-dissipation central-upwind numerical flux and a second-order piecewise linear reconstruction with the slopes computed using an overcompressive SBM limiter. Elsewhere, we use an alternative weighted essentially non-oscillatory (A-WENO) framework with the central-upwind finite-volume numerical fluxes and either unlimited (in smooth regions) or Ai-WENO-Z (in the nonsmooth regions away from contact discontinuities) fifth-order interpolation. Numerical results for the one- and two-dimensional compressible Euler equations show that the proposed adaptive method improves both the computational efficiency and resolution of complex flow features compared with the non-adaptive fifth-order A-WENO scheme.

math.NA↗

arXiv2025

Dual Formulation Finite-Volume Methods on Overlapping Meshes for Hyperbolic Conservation Laws

In this work, we introduce new second-order schemes for one- and two-dimensional hyperbolic systems of conservation laws. Following an approach recently proposed in [{\sc R. Abgrall}, Commun. Appl. Math. Comput., 5 (2023), pp. 370--402], we consider two different formulations of the studied system (the original conservative formulation and a primitive one containing nonconservative products), and discretize them on overlapping staggered meshes using two different numerical schemes. The novelty of our approach is twofold. First, we introduce an original paradigm making use of overlapping finite-volume (FV) meshes over which cell averages of conservative and primitive variables are evolved using semi-discrete FV methods: The nonconservative system is discretized by a path-conservative central-upwind scheme, and its solution is used to evaluate very simple numerical fluxes for the discretization of the original conservative system. Second, to ensure the nonlinear stability of the resulting method, we design a post-processing, which also guarantees a conservative coupling between the two sets of variables. We test the proposed semi-discrete dual formulation finite-volume methods on several benchmarks for the Euler equations of gas dynamics.

math.NA↗

arXiv2025

The Evolution of Pointwise Statistics in Hyperbolic Equations with Random Data

We consider one-dimensional hyperbolic PDEs, linear and nonlinear, with random initial data. Our focus is the {\em pointwise statistics,} i.e., the probability measure of the solution at any fixed point in space and time. For linear hyperbolic equations, the probability density function (PDF) of these statistics satisfies the same linear PDE. For nonlinear hyperbolic PDEs, we derive a linear transport equation for the cumulative distribution function (CDF) and a nonlocal linear PDE for the PDF. Both results are valid only as long as no shocks have formed, a limitation which is inherent to the problem, as demonstrated by a counterexample. For systems of linear hyperbolic equations, we introduce the multi-point statistics and derive their evolution equations. In all of the settings we consider, the resulting PDEs for the statistics are of practical significance: they enable efficient evaluation of the random dynamics, without requiring an ensemble of solutions of the underlying PDE, and their cost is not affected by the dimension of the random parameter space. Additionally, the evolution equations for the statistics lead to a priori statistical error bounds for Monte Carlo methods (in particular, Kernel Density Estimators) when applied to hyperbolic PDEs with random data.

math.AP↗

arXiv2025

Numerical Study of Random Kelvin-Helmholtz Instability

In this paper, we study random dissipative weak solutions of the compressible Euler equations in the Kelvin-Helmholtz (KH) instability. Motivated by the fact that weak entropy solutions are not unique and can be viewed as inviscid limits of Navier-Stokes flows, we take a statistical approach following ideas from turbulence theory. Our aim is to identify solution features that remain consistent across different realizations and mesh resolutions. For this purpose, we compute stable numerical solutions using a stochastic collocation method implemented with the help of a fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme and seventh-order central weighted essentially non-oscillatory (CWENO) interpolation in the random space. The obtained solutions are averaged over several embedded uniform grids, resulting in Ces\'aro averages, which are studied using stochastic tools. The analysis includes Reynolds stress and energy defects, probability density functions of averaged quantities, and reduced-order representations using proper orthogonal decomposition. The presented numerical experiments illustrate that random KH instabilities can be systematically described using statistical methods, averaging, and reduced-order modeling, providing a robust methodology for capturing the complex and chaotic dynamics of inviscid compressible flows.

math.NA↗

arXiv2025

Uncertainty Quantification in Forward Problems: Balancing Accuracy and Robustness Using CWENO Interpolations

In this paper, we study uncertainty quantification (UQ) in forward problems. Our objective is to construct accurate and robust surrogate models by incorporating the seventh-order central weighted essentially non-oscillatory (CWENO7) scheme into the stochastic collocation framework. A key focus is on mitigating the oscillatory behavior often encountered in traditional spectral methods while retaining high-order accuracy in smooth regions. We present a systematic comparison between CWENO7-based and generalized polynomial chaos (gPC)-based approaches. Although gPC methods achieve spectral convergence, they are prone to Gibbs-type oscillations in nonsmooth settings. By contrast, CWENO7 utilizes local stencils to achieve a balance: non-oscillatory behavior near discontinuities and high-order convergence in smooth regions. To validate the approach, we conduct numerical experiments on a range of one- and two-dimensional smooth and nonsmooth problems, including shallow water equations with random inputs. The results demonstrate that CWENO7 interpolation provides accurate estimates of probability density functions, mean values, and standard deviations, particularly in regimes where gPC expansions exhibit strong oscillations. Furthermore, computational tests confirm that CWENO7 interpolation is efficient and scalable, establishing it as a reliable alternative to conventional stochastic collocation techniques for UQ in the presence of discontinuities.

math.NA↗

arXiv2025

Modulation of the Monokinetic Limit for Models of Collective Dynamics

In this work, we perform modulation analysis of monokinetic limits from the kinetic Cucker- Smale model to the pressureless Euler alignment system. Two regimes are considered -- a strong Fokker- Planck force with vanishing noise and Knudsen number, and a pure noiseless Vlasov scheme. In the former case, we demonstrate convergence of the modulated profile to the standard Gaussian distribution, while in the latter case, the distribution converges to a profile satisfying an explicit transport equation along limiting characteristics.

math.AP↗

arXiv2025

New Smoothness Indicator Within an Active Flux Framework

In this work, we introduce a new smoothness indicator (SI), which is capable of detecting ``rough'' parts of the solutions computed by active flux (AF) methods for hyperbolic (systems of) conservation laws. The new SI is based on measuring the difference between the two sets of solutions (either cell averages and point values or cell averages on overlapping grids) evolved at each time step of AF methods. The key idea in the derivation of the new SI is that in the ``rough'' parts of the evolved solutions, the difference is ${\cal O}(1)$, while in the smooth areas, it is proportional to the order of the underlying AF method. The performance of the new SI, that is, its ability to automatically and robustly detect ``rough'' parts of the computed solutions, is illustrated on several numerical examples, in which the one-dimensional Euler equations of gas dynamics are numerically solved by a recently introduced semi-discrete finite-volume AF method on overlapping grids.

math.NA↗

arXiv2025

CWENO Interpolation for Non-Oscillatory Stochastic Collocation in Uncertainty Quantification Problems

Uncertainty quantification (UQ) in mathematical models is essential for accurately predicting system behavior under variability. This study provides guidance on method selection for reliable UQ across varied functional behaviors in engineering applications. Specifically, we compare several interpolation and approximation methods within a stochastic collocation (SC) framework, namely: generalized polynomial chaos (gPC), B-splines, shape-preserving (SP) splines, and central weighted essentially nonoscillatory (CWENO) interpolation, to reconstruct probability density functions (PDFs) and estimate statistical moments. These methods are assessed for both smooth and discontinuous functions, as well as for the solution of the 1-D Euler and shallow water equations. While gPC and interpolation B-splines perform well with smooth data, they produce oscillations near discontinuities. Approximation B-splines and SP splines, while avoiding oscillations, converge more slowly. In contrast, CWENO interpolation demonstrates high robustness, effectively capturing sharp gradients without oscillations, making it suitable for complex, discontinuous data. Overall, CWENO interpolation emerges as a versatile and effective approach for SC, particularly in handling discontinuities in UQ.

math.NA↗

arXiv2024

Spline-Based Stochastic Collocation Methods for Uncertainty Quantification in Nonlinear Hyperbolic PDEs

In this paper, we study the stochastic collocation (SC) methods for uncertainty quantification (UQ) in hyperbolic systems of nonlinear partial differential equations (PDEs). In these methods, the underlying PDEs are numerically solved at a set of collocation points in random space. A standard SC approach is based on a generalized polynomial chaos (gPC) expansion, which relies on choosing the collocation points based on the prescribed probability distribution and approximating the computed solution by a linear combination of orthogonal polynomials in the random variable. We demonstrate that this approach struggles to accurately capture discontinuous solutions, often leading to oscillations (Gibbs phenomenon) that deviate significantly from the physical solutions. We explore alternative SC methods, in which one can choose an arbitrary set of collocation points and employ shape-preserving splines to interpolate the solution in a random space. Our study demonstrates the effectiveness of spline-based collocation in accurately capturing and assessing uncertainties while suppressing oscillations. We illustrate the superiority of the spline-based collocation on two numerical examples, including the inviscid Burgers and shallow water equations.

math.NA↗

arXiv2024

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.

math.NA↗

arXiv2024

A Hybrid Finite-Difference-Particle Method for Chemotaxis Models

Chemotaxis systems play a crucial role in modeling the dynamics of bacterial and cellular behaviors, including propagation, aggregation, and pattern formation, all under the influence of chemical signals. One notable characteristic of these systems is their ability to simulate concentration phenomena, where cell density undergoes rapid growth near specific concentration points or along certain curves. Such growth can result in singular, spiky structures and lead to finite-time blowups. Our investigation focuses on the dynamics of the Patlak-Keller-Segel chemotaxis system and its two-species extensions. In the latter case, different species may exhibit distinct chemotactic sensitivities, giving rise to very different rates of cell density growth. Such a situation may be extremely challenging for numerical methods as they may fail to accurately capture the blowup of the slower-growing species mainly due to excessive numerical dissipation. In this paper, we propose a hybrid finite-difference-particle (FDP) method, in which a particle method is used to solve the chemotaxis equation(s), while finite difference schemes are employed to solve the chemoattractant equation. Thanks to the low-dissipation nature of the particle method, the proposed hybrid scheme is particularly adept at capturing the blowup behaviors in both one- and two-species cases. The proposed hybrid FDP methods are tested on a series of challenging examples, and the obtained numerical results demonstrate that our hybrid method can provide sharp resolution of the singular structures even with a relatively small number of particles. Moreover, in the two-species case, our method adeptly captures the blowing-up solution for the component with lower chemotactic sensitivity, a feature not observed in other works.

math.NA↗

arXiv2023

Divergence-Free Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Rotating Shallow Water Magnetohydrodynamics

We develop a new second-order flux globalization based path-conservative central-upwind (PCCU) scheme for rotating shallow water magnetohydrodynamic equations. The new scheme is designed not only to maintain the divergence-free constraint of the magnetic field at the discrete level but also to satisfy the well-balanced (WB) property by exactly preserving some physically relevant steady states of the underlying system. The locally divergence-free constraint of the magnetic field is enforced by following the method recently introduced in [A. Chertock, A. Kurganov, M. Redle, and K. Wu, ArXiv preprint (2022), arXiv:2212.02682]: we consider a Godunov-Powell modified version of the studied system, introduce additional equations by spatially differentiating the magnetic field equations, and modify the reconstruction procedures for magnetic field variables. The WB property is ensured by implementing a flux globalization approach within the PCCU scheme, leading to a method capable of preserving both still- and moving-water equilibria exactly. In addition to provably achieving both the WB and divergence-free properties, the new method is implemented on an unstaggered grid and does not require any (approximate) Riemann problem solvers. The performance of the proposed method is demonstrated in several numerical experiments that confirm the lack of spurious oscillations, robustness, and high resolution of the obtained results.

math.NA↗

arXiv2023

Denoising Convolution Algorithms and Applications to SAR Signal Processing

Convolutions are one of the most important operations in signal processing. They often involve large arrays and require significant computing time. Moreover, in practice, the signal data to be processed by convolution may be corrupted by noise. In this paper, we introduce a new method for computing the convolutions in the quantized tensor train (QTT) format and removing noise from data using the QTT decomposition. We demonstrate the performance of our method using a common mathematical model for synthetic aperture radar (SAR) processing that involves a sinc kernel and present the entire cost of decomposing the original data array, computing the convolutions, and then reformatting the data back into full arrays.

math.NA↗

arXiv2022

A New Locally Divergence-Free Path-Conservative Central-Upwind Scheme for Ideal and Shallow Water Magnetohydrodynamics

We develop a new second-order unstaggered path-conservative central-upwind (PCCU) scheme for ideal and shallow water magnetohydrodynamics (MHD) equations. The new scheme possesses several important properties: it locally preserves the divergence-free constraint, it does not rely on any (approximate) Riemann problem solver, and it robustly produces high-resolution and non-oscillatory results. The derivation of the scheme is based on the Godunov-Powell nonconservative modifications of the studied MHD systems. The local divergence-free property is enforced by augmenting the modified systems with the evolution equations for the corresponding derivatives of the magnetic field components. These derivatives are then used to design a special piecewise linear reconstruction of the magnetic field, which guarantees a non-oscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization accounts for the jump of the nonconservative product terms across cell interfaces, thereby ensuring stability. We test the proposed PCCU scheme on several benchmarks for both ideal and shallow water MHD systems. The obtained numerical results illustrate the performance of the new scheme, its robustness, and its ability not only to achieve high resolution, but also preserve the positivity of computed quantities such as density, pressure, and water depth.

math.NA↗

arXiv2022

Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator

We develop new adaptive alternative weighted essentially non-oscillatory (A-WENO) schemes for hyperbolic systems of conservation laws. The new schemes employ the recently proposed local characteristic decomposition based central-upwind numerical fluxes, the three-stage third-order strong stability preserving Runge-Kutta time integrator, and the fifth-order WENO-Z interpolation. The adaptive strategy is implemented by applying the limited interpolation only in the parts of the computational domain where the solution is identified as ``rough'' with the help of a smoothness indicator. We develop and use a new simple and robust local smoothness indicator (LSI), which is applied to the solutions computed at each of the three stages of the ODE solver. The new LSI and adaptive A-WENO schemes are tested on the Euler equations of gas dynamics. We implement the proposed LSI using the pressure, which remains smooth at contact discontinuities, while our goal is to detect other ``rough'' areas and apply the limited interpolation mostly in the neighborhoods of the shock waves. We demonstrate that the new adaptive schemes are highly accurate, non-oscillatory, and robust. They outperform their fully limited counterparts (the A-WENO schemes with the same numerical fluxes and ODE solver but with the WENO-Z interpolation employed everywhere) while being less computationally expensive.

math.NA↗

arXiv2022

A Diffuse-Domain Based Numerical Method for a Chemotaxis-Fluid Model

In this paper, we consider a coupled chemotaxis-fluid system that models self-organized collective behavior of oxytactic bacteria in a sessile drop. This model describes the biological chemotaxis phenomenon in the fluid environment and couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier-Stokes equations subject to a gravitational force, which is proportional to the relative surplus of the cell density compared to the water density. We develop a new positivity preserving and high-resolution method for the studied chemotaxis-fluid system. Our method is based on the diffuse-domain approach, which we use to derive a new chemotaxis-fluid diffuse-domain (cf-DD) model for simulating bioconvection in complex geometries. The drop domain is imbedded into a larger rectangular domain, and the original boundary is replaced by a diffuse interface with finite thickness. The original chemotaxis-fluid system is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the physical interface. We show that the cf-DD model converges to the chemotaxis-fluid model asymptotically as the width of the diffuse interface shrinks to zero. We numerically solve the resulting cf-DD system by a second-order hybrid finite-volume finite-difference method and demonstrate the performance of the proposed approach on a number of numerical experiments that showcase several interesting chemotactic phenomena in sessile drops of different shapes, where the bacterial patterns depend on the droplet geometries.

math.NA↗