Conservative Finite Volume Research Articles

A Liouville optimal control framework in prostate cancer

In this work we present a new stochastic framework for obtaining optimal treatment regimes in prostate cancer. We model the realistic scenario of randomized clinical trials for incorporating randomness related to interaction between a prostate cancer cell and androgen cell quota, due to cancer heterogeneities, across different patients in a given group, using a Liouville partial differential equation. We then solve two optimization problems: one for determining the model parameters to fit the measured data and the second to determine the optimal androgen deprivation therapy. The optimization problems are implemented using a positive, stable, and conservative finite volume solver for the Liouville equations and the projected non-linear conjugate gradient method. Several numerical results, including comparison with ordinary differential equations modeling framework, demonstrate the robustness and accuracy of our proposed framework to obtain optimal treatment regimes in real time.

A Semi-implicit Finite Volume Scheme for Incompressible Two-Phase Flows

This paper presents a mass and momentum conservative semi-implicit finite volume (FV) scheme for complex non-hydrostatic free surface flows, interacting with moving solid obstacles. A simplified incompressible Baer-Nunziato type model is considered for two-phase flows containing a liquid phase, a solid phase, and the surrounding void. According to the so-called diffuse interface approach, the different phases and consequently the void are described by means of a scalar volume fraction function for each phase. In our numerical scheme, the dynamics of the liquid phase and the motion of the solid are decoupled. The solid is assumed to be a moving rigid body, whose motion is prescribed. Only after the advection of the solid volume fraction, the dynamics of the liquid phase is considered. As usual in semi-implicit schemes, we employ staggered Cartesian control volumes and treat the nonlinear convective terms explicitly, while the pressure terms are treated implicitly. The non-conservative products arising in the transport equation for the solid volume fraction are treated by a path-conservative approach. The resulting semi-implicit FV discretization of the mass and momentum equations leads to a mildly nonlinear system for the pressure which can be efficiently solved with a nested Newton-type technique. The time step size is only limited by the velocities of the two phases contained in the domain, and not by the gravity wave speed nor by the stiff algebraic relaxation source term, which requires an implicit discretization. The resulting semi-implicit algorithm is first validated on a set of classical incompressible Navier-Stokes test problems and later also adds a fixed and moving solid phase.

An upwind-mixed volume element method on changing meshes for compressible miscible displacement problem

Numerical simulation of oil-water miscible displacement is discussed in this paper, and a compressible problem of energy mathematics is solved potentially. The mathematical model defined by a nonlinear system includes mainly two partial differential equations (PDEs): a parabolic equation for the pressure and a convection-diffusion equation for the saturation. The pressure is obtained by a conservative mixed finite volume element method (MFVE). The computational accuracy is improved for Darcy velocity. A conservative upwind mixed finite volume element method (UMFVE) is applied to compute the saturation on changing meshes. The diffusion is discretized by a mixed finite volume element method, and the convection is computed by upwind differences. The upwind method can solve convection-dominated diffusion equations accurately and avoids numerical dispersion and nonphysical oscillation. The saturation and the adjoint vector function are obtained simultaneously. An optimal order error estimates is obtained. Finally, numerical examples are provided to show the accuracy, efficiency and possible applications.

A Lagrangian method for slurry flow modeling in hydraulic fractures

We developed a new numerical simulator of slurry flow in hydraulic fractures. It is based on a continuum model developed under the assumptions of lubrication theory. We consider slurry containing fluid and proppant components. Proppant particles settle under gravity and bridge in the flow domain. Fluid fluxes are calculated from an elliptic boundary value problem for given fracture geometry and sources/sinks distribution. We solve it using a conservative finite volume method and then apply empirical closure relations to get the velocity field for solids. The transport of slurry components is described by nonlinear hyperbolic equations with fluxes depending on proppant concentration and particle size/fracture width ratio. These equations admit solutions with concentration shock waves and rarefaction fans. They must be represented accurately in the computational algorithm. We utilize an Eulerian approach for the discretization of nonlinear proppant fluxes. After getting the fluxes on the interfaces between Eulerian grid cells we follow a Lagrangian approach to simulate the proppant transport along the given velocity field. In this study we discuss key implementation aspects: time step selection, regularization, fluxes discretization, seeding, reseeding, and movement of Lagrangian particles. We verify the simulator on benchmark solutions focused on advection, batch proppant settling, slurry slumping and fluid displacement by proppant.

A conservative finite volume method for incompressible two-phase flows on unstructured meshes

A robust finite volume framework for solving incompressible two-phase flow problems based on an explicit dual-time stepping based artificial compressibility formulation has been developed by Parameswaran and Mandal (2019) for structured meshes. This artificial compressibility framework is extended to unstructured meshes in the present paper. In order to further enhance the applicability to real life engineering problems, the surface tension effect is also modeled here. To deal with arbitrary unstructured mesh topology, least square and Green-Gauss integral based methods are used. The efficacy of the proposed method is demonstrated using a set of scalar advection based test problems and a set of standard incompressible two-phase flow test problems involving gravitational, viscous and surface tension forces. The numerical results show improvement in accuracy compared to the results of the structured mesh formulation.

A learned conservative semi-Lagrangian finite volume scheme for transport simulations

Semi-Lagrangian (SL) schemes are known as a major numerical tool for solving transport equations with many advantages and have been widely deployed in the fields of computational fluid dynamics, plasma physics modeling, numerical weather prediction, among others. In this work, we develop a novel machine learning-assisted approach to accelerate the conventional SL finite volume (FV) schemes. The proposed scheme avoids the expensive tracking of upstream cells but attempts to learn the SL discretization from the data by incorporating specific inductive biases in the neural network, significantly simplifying the algorithm implementation and leading to improved efficiency. In addition, the method delivers sharp shock transitions and a level of accuracy that would typically require a much finer grid with traditional transport solvers. Numerical tests demonstrate the effectiveness and efficiency of the proposed method.

An efficient implementation of a conservative finite volume scheme with constant and linear reconstructions for solving the coagulation equation

Population balances provide an economic means of describing dense particulate phases with particle-particle interactions. Here, we focus on particle coagulation and address the evaluation of the corresponding integral source terms within the scope of a conservative finite volume formulation. Based on one kernel evaluation for every pair of parent cells, efficient analytical formulas are derived that obviate the explicit decomposition of the integration domains into elementary geometrical shapes and are valid for arbitrary volume-grids. Considering the volume-weighted cell averages as degrees of freedom, the formulas are presented for both cell-wise constant and linear reconstructions of the particle volume distribution. We find the latter to be effective at mitigating the unphysically heavy tails that are characteristic of solutions obtained with piecewise constant reconstructions. For Brownian coagulation, the computational expense is mainly caused by the kernel evaluations, while the cost of the parent-daughter pairing and integration algorithm is almost negligibly small.

Non-linear Eulerian hydrodynamics of dark energy: Riemann problem and finite volume schemes

Upcoming large-scale structure surveys can shed new light on the properties of dark energy. In particular, if dark energy is a dynamical component, it must have spatial perturbations. Their behaviour is regulated by the speed of sound parameter, which is currently unconstrained. In this work, we present the numerical methods that will allow to perform cosmological simulations of inhomogeneous dark energy scenarios where the speed of sound is small and non-vanishing. We treat the dark energy component as an effective fluid and build upon established numerical methods for hydrodynamics to construct a numerical solution of the effective continuity and Euler equations. In particular, we develop conservative finite volume schemes that rely on the solution of the Riemann problem, which we provide here in both exact and approximate forms for the case of a dark energy fluid.

Algebraic flux correction finite element method with semi-implicit time stepping for solute transport in fractured porous media

This work is concerned with the numerical modeling of the Darcy flow and solute transport in fractured porous media for which the fractures are modeled as interfaces of codimension one. The hybrid-dimensional flow and transport problems are discretizaed by a lumped piece-wise linear finite element method, combined with the algebraic correction of the convective fluxes. The resulting transport discretization can be interpreted as a conservative finite volume scheme that satisfies the discrete maximum principle, while introducing a very limited amount of numerical diffusion. In the context of fractured porous media flow the CFL number may vary by several order of magnitude, which makes explicit time stepping unfeasible. To cope with this difficulty we propose an adaptive semi-implicit time stepping strategy that reduces to the low order linear implicit discretization in the high CFL regions that include, but may not be limited to the fracture network. The performance of the fully explicit and semi-implicit variants of the method are investigated through the numerical experiment.

A conservative finite volume method for the population balance equation with aggregation, fragmentation, nucleation and growth

In the present paper, we present a method for solving the population balance equation (PBE) with the complete range of kinetic processes included: namely aggregation, fragmentation, nucleation and growth. The method is based on the finite volume scheme and features guaranteed conservation of the first moment by construction, accurate prediction of the size distribution, applicability to an arbitrary non-uniform grid, robustness and computational efficiency which is instrumental for coupling with computational fluid dynamics (CFD). The treatment of aggregation is based on the previous work by Liu and Rigopoulos (2019). An analysis of the aggregation terms in the PBE is made, and the source of conservation error in finite element/volume methods is elucidated. It is subsequently shown how this error is overcome in the present method via a coordinate transformation applied to the aggregation birth double integral resulting from the application of the finite volume method. The contributions to the birth term are delineated and their corresponding death fluxes identified. An aggregation map is then constructed for mapping birth and death fluxes, thus allowing the finite volume method to operate in terms of fluxes and achieve conservation of mass. The method is then extended to fragmentation, for which a map is also constructed to represent the birth and death fluxes. In the implementation, the aggregation and fragmentation maps are pre-tabulated to allow fast computation. It is also shown how the method can be coupled with a total variation diminishing (TVD) scheme for the treatment of growth with minimal numerical diffusion. The method is validated with a number of test cases including analytical solutions and numerical solutions of the discrete PBE for aggregation (theoretical and free molecule/Brownian kernels), fragmentation, aggregation-fragmentation and aggregation-growth. In all cases, the method produces very accurate results, while also being computationally efficient due to the pre-tabulation of the maps and the simplicity of the algorithm carried out per time step.

Exact solution for Riemann problems of the shear shallow water model

The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks, rarefactions, shear and contact waves. The notion of weak solution is based on a path but the choice of the correct path is not known for this problem. In this paper, we construct exact solution for the Riemann problem assuming a linear path in the space of conserved variables, which is also used in approximate Riemann solvers. We compare the exact solutions with those obtained from a path conservative finite volume scheme on some representative test cases.

Conservative finite volume scheme for first-order viscous relativistic hydrodynamics

We present the first conservative finite volume numerical scheme for the causal, stable relativistic Navier-Stokes equations developed by Bemfica, Disconzi, Noronha, and Kovtun (BDNK). BDNK theory has arisen very recently as a promising means of incorporating entropy-generating effects (viscosity, heat conduction) into relativistic fluid models, appearing as a possible alternative to the so-called M\"uller-Israel-Stewart (MIS) theory successfully used to model quark-gluon plasma. The major difference between the two lies in the structure of the system of partial differential equations (PDEs): BDNK theory only has a set of conservation laws, whereas MIS also includes a set of evolution equations for its dissipative degrees of freedom. The simpler structure of the BDNK PDEs in this respect allows for rigorous proofs of stability, causality, and hyperbolicity in full generality which have as yet been impossible for MIS. To capitalize on these advantages, we present the first fully conservative multidimensional fluid solver for the BDNK equations suitable for physical applications. The scheme includes a flux-conservative discretization, nonoscillatory reconstruction, and a central-upwind numerical flux and is designed to smoothly transition to a high-resolution shock-capturing perfect fluid solver in the inviscid limit. We assess the robustness of our new method in a series of flat-spacetime tests for a conformal fluid and provide a detailed comparison with previous approaches of Pandya and Pretorius [Phys. Rev. D 104, 023015 (2021)].

A Three-dimensional Multi-species Flow Solver for the Euler Equations Combined with a Stiffened Gas Equation of State

Although numerical simulation in fluid mechanics is undergoing a significant development due to the dazzling evolution of computing means, complex physical phenomena, such as multidimensional viscous effects in turbomachinery and cavitation, remain mysterious and attract the curiosity of several researchers. Highresolution shock captures are often obtained by the WENO family of schemes, except that in problems that depend on discontinuities and shocks, an appearance of numerical oscillations weakens its ability to provide adequate captures. The use of the characteristic construction methods prevents this type of oscillation. The present paper contributes to the numerical resolution of multi-species flows of viscous, compressible, or incompressible fluids with shocks and discontinuities. The proposed numerical model can handle various configurations with a unique method based on a conservative and consistent threedimensional finite volume scheme with an aligned mesh. The system of equations is a set of Euler equations coupled with a two-parameters generalized state equation of state in three-dimensional Cartesian coordinates. This system is solved using a Roe type approximate Riemann solver, and second-order precision is obtained using limiters. The obtained numerical results maintain a nonoscillatory flow near the discontinuities, which makes the method satisfactory and shows its accuracy and robustness in different cases.

Modelling of soot formation and aggregation in turbulent flows with the LES-PBE-PDF approach and a conservative sectional method

The present paper presents a comprehensive approach for modelling soot formation in turbulent flows that incorporates a recently developed conservative finite volume sectional method for the solution of the population balance equation (PBE) into the Large Eddy Simulation - Probability Density Function (LES-PDF) framework for modelling turbulent reacting flows. The approach thus predicts the particle size distribution of soot and accounts for the complete set of aerosol dynamic processes (nucleation, surface growth, oxidation, condensation and aggregation) as well as for the interaction between turbulence, chemistry and soot. We employ a combination of gas-phase chemistry, soot kinetics and aerosol dynamics recently used by the authors in the context of a laminar co-flow diffusion flame and maintain the same kinetics without adjustments in order to focus on the modelling issues in turbulent sooting flame simulations. The results show that several features such as the location of the soot forming zones and the qualitative profiles of soot volume fraction and species concentrations can be predicted reasonably well, but there is a considerable underprediction in the magnitude of soot volume fraction. The possible reasons for this underprediction are discussed in the context of the aforementioned laminar flame simulation and the previous validation of the components of our methodology. The CPU time analysis shows that the solution of the PBE by the conservative sectional method and the associated transport of the discretised number densities account for less than 25% of the total CPU time, indicating that its incorporation in LES-PDF simulations of turbulent combustion is computationally feasible.

A fourth-order conservative semi-Lagrangian finite volume WENO scheme without operator splitting for kinetic and fluid simulations

In this paper, we present a fourth-order conservative semi-Lagrangian (SL) finite volume (FV) weighted essentially non-oscillatory (WENO) scheme without operator splitting for two-dimensional linear transport equations with applications in kinetic models including the nonlinear Vlasov–Poisson system, the guiding center Vlasov model and the incompressible Euler equation in the vorticity-stream function formulation. To achieve fourth-order accuracy in space, two main ingredients are proposed in the SL FV formulation. Firstly, we introduce a so-called cubic-curved quadrilateral upstream cell and applying an efficient clipping method to evaluate integrals on upstream cells. Secondly, we construct a new WENO reconstruction operator, which recovers a P3 polynomial from neighboring cell averages. Mass conservation is accomplished with the mass conservative nature of the reconstruction operator and the SL formulation. A positivity-preserving limiter is applied to maintain the positivity of the numerical solution wherever appropriate. For nonlinear kinetic models, the SL scheme is coupled with a fourth-order Runge–Kutta exponential integrator for high-order temporal accuracy. Extensive benchmarks are tested to verify the designed properties.

Least-squares ReLU neural network (LSNN) method for scalar nonlinear hyperbolic conservation law

In [7], we introduced the least-squares ReLU neural network (LSNN) method for solving the linear advection-reaction problem with discontinuous solution and showed that the method outperforms mesh-based numerical methods in terms of the number of degrees of freedom. This paper studies the LSNN method for scalar nonlinear hyperbolic conservation law. The method is a discretization of an equivalent least-squares (LS) formulation in the set of neural network functions with the ReLU activation function. Evaluation of the LS functional is done by using numerical integration and conservative finite volume scheme. Numerical results of some test problems show that the method is capable of approximating the discontinuous interface of the underlying problem automatically through the free breaking lines of the ReLU neural network. Moreover, the method does not exhibit the common Gibbs phenomena along the discontinuous interface.

Численное моделирование процесса кристаллизации трехкомпонентного раствора на подвижной сетке

The crystallization process of ternary alloy in the rectangular growth chamber is considered. The mathematical model accounts for crystallization interface movement, convective mass transfer in the solution, diffusion mass transfer in crystal, equilibrium conditions at the phase transition interface. The conservative finite volume scheme is constructed on the boundary-fitted moving grid. The discrete counterparts of conservation laws for kinetic energy and mass are satisfied in the numerical model. The effect of growth chamber dimensions on the crystal composition is studied numerically.

A 3D cell-centered ADER MOOD Finite Volume method for solving updated Lagrangian hyperelasticity on unstructured grids

In this paper, we present a conservative cell-centered Lagrangian Finite Volume scheme for solving the hyperelasticity equations on unstructured multidimensional grids. The starting point of the present approach is the cell-centered FV discretization named EUCCLHYD and introduced in the context of Lagrangian hydrodynamics. Here, it is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction. The ADER (Arbitrary high order schemes using DERivatives) approach is adopted to obtain second-order of accuracy in time. This strategy has been successfully tested in a hydrodynamics context and the present work aims at extending it to the case of hyperelasticity. Here, the hyperelasticity equations are written in the updated Lagrangian framework and the dedicated Lagrangian numerical scheme is derived in terms of nodal solver, Geometrical Conservation Law (GCL) compliance, subcell forces and compatible discretization. The Lagrangian numerical method is implemented in 3D under MPI parallelization framework allowing to handle genuinely large meshes. A relatively large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of the solution where appropriate. Pure elastic neo-Hookean and non-linear materials are considered for our benchmark test problems in 2D and 3D. These test cases feature material bending, impact, compression, non-linear deformation and further bouncing/detaching motions.

Mortar-based Entropy-Stable Discontinuous Galerkin Methods on Non-conforming Quadrilateral and Hexahedral Meshes

High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points (Chan et al. in SIAM J Sci Comput 41(5):A2938–A2966, 2019) . In this work, we extend entropy-stable Gauss collocation schemes to non-conforming meshes. Entropy-stable DG schemes require computing entropy conservative numerical fluxes between volume and surface quadrature nodes. On conforming tensor product meshes where volume and surface nodes are aligned, flux evaluations are required only between “lines” of nodes. However, on non-conforming meshes, volume and surface nodes are no longer aligned, resulting in a larger number of flux evaluations. We reduce this expense by introducing an entropy-stable mortar-based treatment of non-conforming interfaces via a face-local correction term, and provide necessary conditions for high-order accuracy. Numerical experiments for the compressible Euler equations in two and three dimensions confirm the stability and accuracy of this approach.

Conservative finite volume method on deforming geometries: The case of protein aggregation in dividing yeast cells

Motivated by experimental observations of asymmetric protein aggregates distributions in dividing yeast cells, we present a conservative finite volume approach for reaction-diffusion systems defined over deforming geometries. The key idea of our approach is to use spatio-temporal control volumes instead of integrating the time-discretized equations in space, as it is common practice. Both our theoretical and computational results demonstrate the convergence of our method and highlight how traditional approaches can lead to inaccurate solutions. We employ this novel approach to investigate the partitioning of protein aggregates in dividing yeast cells, leveraging the flexibility of the level set method to construct realistic biological geometries. Using a simple reaction-diffusion model, we find that spatial heterogeneity in yeast cells during division can alone create asymmetries in the concentration of protein aggregates. Moreover, we find that obstructing intracellular entities, such as nuclei or insoluble protein compartments, amplify these asymmetries, suggesting that they may play an essential role in regulating molecular partitioning. Beyond these findings, our results illustrate the flexibility of our approach and its potential to design realistic predictive tools to explore intracellular bio-mechanisms.