A Positive Finite-Difference Advection Scheme (2023)

Large-eddy simulation (LES) is used to systematically analyse the impacts of trees on air quality in idealised street canyons. The LES tree model includes radiation, transpiration, drag and deposition effects. The superposition of background concentrations and local emissions is used to construct realistic urban scenarios for fine particulate matter (PM2.5) and nitrogen oxides (NO_x). Both neutral and convective atmospheric conditions are considered to assess the importance of buoyancy effects and the role of tree shading and transpiration. Tree impact on local air quality is shown to be driven by the balance between the rate at which they actively remove pollutants from the air (deposition) and the way in which they alter the transport of pollutants within and out of the street canyon (dispersion). For pollutant species or street types where the concentration field is dominated by background levels (such as PM2.5), deposition will generally dominate and thus local air quality will improve. For pollutants and street types where local emission sources dominate (e.g. NO_x on a busy road), the dispersion effects of trees become more prominent and can lead to elevated concentrations where mixing or exchange is significantly inhibited. Mixing in the convective simulation is more vigorous than in the neutral simulation which results in substantial differences in in-canyon flow fields and exchange velocities, highlighting the importance of incorporating thermal effects when studying urban trees. Increased residency times, and thus deposition, under neutral conditions suggest that trees can have amplified effects under conditions conducive of poor air quality. For the cases considered, trees largely act to improve air quality with the exception of localised hotspots. The competing effects of trees — specifically deposition versus altered exchange with the atmosphere — are also incorporated in a simple integral model that predicts whether or not the air quality will improve. The model matches well with LES predictions for both PM2.5 and NO_x and can serve as a simple tool for urban design purposes.

In preparation of the study of liquefied natural gas (LNG) sloshing in ships and vehicles, we model and numerically analyze compressible two-fluid flow. We consider a five-equation two-fluid flow model, assuming velocity and pressure continuity across two-fluid interfaces, with a separate equation to track the interfaces. The system of partial differential equations is hyperbolic and quasi-conservative. It is discretized in space with a tailor-made third-order accurate finite-volume method, employing an HLLC approximate Riemann solver. The third-order accuracy is obtained through spatial reconstruction with a limiter function, for which a novel formulation is presented. The non-homogeneous term is handled in a way consistent with the HLLC treatment of the convection operator. We study the one-dimensional case of a liquid column impacting onto a gas pocket entrapped at a solid wall. It mimics the impact of a breaking wave in an LNG containment system, where a gas pocket is entrapped at the tank wall below the wave crest. Furthermore, the impact of a shock wave on a gas bubble containing the heavy gas R22, immersed in air, is simulated in two dimensions and compared with experimental results. The numerical scheme is shown to be higher-order accurate in space and capable of capturing the important characteristics of compressible two-fluid flow.

We consider spectral discretizations of hyperbolic problems on unbounded domains using Laguerre basis functions. Taking as model problem the scalar advection equation, we perform a comprehensive stability analysis that includes strong collocation formulations, nodal and modal weak formulations, with either inflow or outflow boundary conditions, using either Gauss–Laguerre or Gauss–Laguerre–Radau quadrature and based on either scaled Laguerre functions or scaled Laguerre polynomials. We show that some of these combinations give rise to intrinsically unstable discretizations, while the combination of scaled Laguerre functions with Gauss–Laguerre–Radau quadrature appears to be stable for both strong and weak formulations. We then show how a modal discretization approach for hyperbolic systems on an unbounded domain can be naturally and seamlessly coupled to a discontinuous finite element discretization on a finite domain. An example of one dimensional hyperbolic system is solved with the proposed domain decomposition technique. The errors obtained with the proposed approach are found to be small, enabling the use of the coupled scheme for the simulation of Rayleigh damping layers in the semi-infinite part. Energy errors and reflection ratios of the scheme in absorbing wavetrains and single Gaussian signals show that a small number of modes in the semi-infinite domain are sufficient to damp the waves. The theoretical insight and numerical results corroborate previous findings by the authors and establish the scaled Laguerre functions-based discretization as a flexible and efficient tool for absorbing layers as well as for the accurate simulation of waves in unbounded regions.

In this work, we propose a positivity-preserving scheme for solving two-dimensional advection–diffusion equations including mixed derivative terms, in order to improve the accuracy of lower-order methods. The solution of these equations, in the absence of mixed derivatives, has been studied in detail, while positivity-preserving schemes for mixed derivative terms have received much less attention. A two-dimensional diffusion equation, for which the analytical solution is known, is solved numerically to show the applicability of the scheme. It is further applied to the Fokker–Planck collision operator in two-dimensional cylindrical coordinates under the assumption of local thermal equilibrium. For a thermal equilibration problem, it is shown that the scheme conserves particle number and energy, while the preservation of positivity is ensured and the steady-state solution is the Maxwellian distribution.

This paper extends the multidimensional positive definite advection transport algorithm (MPDATA) to third-order accuracy for temporally and spatially varying flows. This is accomplished by identifying the leading truncation error of the standard second-order MPDATA, performing the Cauchy–Kowalevski procedure to express it in a spatial form and compensating its discrete representation—much in the same way as the standard MPDATA corrects the first-order accurate upwind scheme. The procedure of deriving the spatial form of the truncation error was automated using a computer algebra system. This enables various options in MPDATA to be included straightforwardly in the third-order scheme, thereby minimising the implementation effort in existing code bases. Following the spirit of MPDATA, the error is compensated using the upwind scheme resulting in a sign-preserving algorithm, and the entire scheme can be formulated using only two upwind passes. Established MPDATA enhancements, such as formulation in generalised curvilinear coordinates, the nonoscillatory option or the infinite-gauge variant, carry over to the fully third-order accurate scheme. A manufactured 3D analytic solution is used to verify the theoretical development and its numerical implementation, whereas global tracer-transport benchmarks demonstrate benefits for chemistry-transport models fundamental to air quality monitoring, forecasting and control. A series of explicitly-inviscid implicit large-eddy simulations of a convective boundary layer and explicitly-viscid simulations of a double shear layer illustrate advantages of the fully third-order-accurate MPDATA for fluid dynamics applications.

Applied Geochemistry, Volume 50, 2014, pp. 74-81

Consideration of the impact of substantial changes in soil temperature or moisture regime on the geochemical forms of radionuclides is important for more accurate assessment of the environmental risk posed by radionuclide migration and potential biological availability, especially in the first months after their release into the environment. This paper presents the results from a study of the influence of cooling, freezing and soil drought on the migration and potential bioavailability of ⁶⁰Co and ¹³⁷Cs in two soils (a fluvisol and a cambisol, according to the World Reference Base for Soil Resources/FAO) from Bulgaria. The changes in the geochemical fractionation of ⁶⁰Co, the exchangeable ¹³⁷Cs and water-soluble forms of both radionuclides were examined under different storage conditions up to 5months after their introduction into the soils in solution form. Freezing or soil drought resulted in a significant increase of the water-soluble forms of ⁶⁰Co in the fluvisol soil, defining higher mobility and potential bioavailability. No influence of the storing conditions on the water-solubility of ⁶⁰Co in the cambisol soil was established. The cooling, freezing and soil drought caused an increase of the exchangeable ¹³⁷Cs in both soils.

Applied Mathematics and Computation, Volume 239, 2014, pp. 74-88

In this paper, a collocation finite difference scheme based on new cubic trigonometric B-spline is developed and analyzed for the numerical solution of a one-dimensional hyperbolic equation (wave equation) with non-local conservation condition. The usual finite difference scheme is used to discretize the time derivative while a cubic trigonometric B-spline is utilized as an interpolation function in the space dimension. The scheme is shown to be unconditionally stable using the von Neumann (Fourier) method. The accuracy of the proposed scheme is tested by using it for several test problems. The numerical results are found to be in good agreement with known exact solutions and with existing schemes in literature.

Journal of Functional Analysis, Volume 266, Issue 4, 2014, pp. 2403-2423

For a continuous field of C^⁎-algebras A, we give a criterion to ensure that the stable rank of A is one. In the particular case of a trivial field this leads to a characterization of stable rank one, completing accomplishments by Nagisa, Osaka and Phillips. Further, for certain continuous fields of C^⁎-algebras, we study when the Cuntz semigroup satisfies the Riesz interpolation property, and we also analyze the structure of its functionals. As an application, we obtain a positive answer to a conjecture posed by Blackadar and Handelman in a variety of situations.

Computer Methods in Applied Mechanics and Engineering, Volume 340, 2018, pp. 500-529

Journal of Mathematical Analysis and Applications, Volume 449, Issue 2, 2017, pp. 1424-1471

In this paper, we study the following nonlinear differential equations of motions of relativistic oscillators with singular potentials where V is a singular potential and p is a 1-periodic function. We will prove the boundedness of all solutions and the existence of infinitely many quasi-periodic solutions via Moser's twist theorem.

Engineering Analysis with Boundary Elements, Volume 78, 2017, pp. 26-30