About
Professor Alina Chertock leads a research group at North Carolina State University focused on developing numerical methods for partial differential equations, scientific computing, multiscale modeling, and uncertainty quantification. She offers graduate student research assistant and postdoctoral fellow positions, providing opportunities to work on exciting research projects and collaborate with experienced professionals in these fields. Her group includes current Ph.D. students and postdoctoral fellows, as well as former students and visiting scholars, reflecting an active and collaborative research environment. Professor Chertock's work contributes to advancing computational techniques and mathematical modeling approaches that address complex scientific and engineering problems.
Research signals
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Research topics
- Geometry
- Mathematics
- Applied mathematics
- Mathematical analysis
- Geology
- Physics
- Algorithm
- Mechanics
Selected publications
arXiv (Cornell University) · 2026-04-28
articleOpen access1st authorCorrespondingWe 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.
New Adaptive Numerical Methods Based on Dual Formulation of Hyperbolic Conservation Laws
ArXiv.org · 2026-01-27
articleOpen access1st authorCorrespondingIn 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.
arXiv (Cornell University) · 2026-04-28
articleOpen access1st authorCorrespondingThe 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 (CU) 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 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.
arXiv (Cornell University) · 2026-04-28
preprintOpen access1st authorCorrespondingWe 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.
arXiv (Cornell University) · 2026-04-28
preprintOpen access1st authorCorrespondingThe 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 (CU) 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 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.
SSRN Electronic Journal · 2026-01-01
preprintOpen access1st authorCorrespondingNew Adaptive Numerical Methods Based on Dual Formulation of Hyperbolic Conservation Laws
Open MIND · 2026-01-27
preprint1st authorCorrespondingIn 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.
Dual formulation finite-volume methods on overlapping meshes for hyperbolic conservation laws
Universität Zürich, ZORA · 2026-02-08
articleOpen accessNew Smoothness Indicator Within an Active Flux Framework
ArXiv.org · 2025-05-01
preprintOpen access1st authorCorrespondingIn 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.
Challenges in Stochastic Galerkin Methods for Nonlinear Hyperbolic Systems with Uncertainty
2025-07-27
book-chapter1st author
Recent grants
NSF · $250k · 2018–2023
NSF · $369k · 2022–2026
NSF · $118k · 2011–2015
Particle Methods for Nonlinear Time-Dependent PDEs
NSF · $161k · 2004–2008
Innovative Numerical Methods for Nonlinear Time-Dependent PDEs
NSF · $272k · 2007–2011
Frequent coauthors
- 78 shared
Alexander Kurganov
- 16 shared
Yaping Wu
- 16 shared
Xuefeng Wang
- 11 shared
Mária Lukáčová-Medviďová
- 9 shared
Г. И. Баренблатт
- 9 shared
Tong Wu
The University of Texas at San Antonio
- 9 shared
A. A. Kurganov
- 9 shared
Pierre Degond
Institut de Mathématiques de Toulouse
Education
- 2013
Ph.D., Mathematics
University of North Carolina at Chapel Hill
- 2010
M.S., Mathematics
University of North Carolina at Chapel Hill
- 2008
B.S., Mathematics
University of North Carolina at Chapel Hill
Awards & honors
- Fellow of the Society for Industrial and Applied Mathematics
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