Hyperbolic Transport Research Articles

Efficient general method for numerically modeling laser pulse propagation, overlap, and lifetime effects in amplifiers

An efficient numerical time-dependent general method is developed to address incoherent pulse overlap and lifetime effects in laser amplifiers. The alternating propagation-population laser energetics method (APPLE) has been validated against a semi-discrete coupled rate equation numerical method (SDRE) and analytic formalisms in bounding cases. APPLE is based on decoupled rates applied to a time-dependent framework where both space-time-dependent populations and pulse energetics are consistently updated in each time step. A significant advantage of APPLE lies in its conceptual simplicity, ease of implementation, and relatively small computational cost. SDRE tracks the populations through coupled rates and uses the method of lines to discretize the hyperbolic partial differential transport equations allowing for use of ordinary differential equation solvers. With reasonably sized mesh, we report both energetic and power pulse shape relative differences on the order of one percent between the models over a large range of initial conditions.

Derivation of a 2D PCCU-AENO method for nonconservative problems

In this paper, we introduce a methodology to design genuinely two-dimensional (2D) second-order well-balanced path-conservative central-upwind (PCCU) schemes. The scheme studies dam-break with high sediment concentration over abrupt moving topography quickly spatially variable even in the presence of the resonance. This study is possible via a 2D hyperbolic sediment transport model (including arbitrarily sloping sediment beds and associated energy and entropy) in new generalized Shallow Water equations derived with associated energy and entropy in this work. We expose some properties of the model and we establish an existence result of global weak solutions. In addition, we show the convergence of a sequence of solutions of the proposed new model. The second-order accuracy of the PCCU scheme is achieved using a new extension AENO (Averaging Essentially Non-Oscillatory) reconstruction developed in the 2D version of this work. We prove that the derived 2D scheme is well-balanced, positivity-preserving and steady states capturing. Several dam break tests are made to show the ability and the superb performances of the proposed numerical modeling. The proposed method can help to address many engineering problems.

Nonlinear thermal analysis of two-dimensional materials with memory

A nonlinear hyperbolic heat transport equation has been proposed based on the Cattaneo model without mechanical effects. We analyze the two-dimensional Maxwell-Cattaneo-Vernotte heat equation in a medium subjected to homogeneous and non-homogeneous boundary conditions and with thermal conductivity and relaxation time linearly dependent on temperature. Since these nonlinearities are essential from an experimental point of view, it is necessary to establish an effective and reliable way to solve the system of partial differential equations and study the behavior of temperature evolution. A numerical scheme of finite differences for the solution of the two-dimensional non-Fourier heat transfer equation is introduced and studied. We also investigate the attributes of the numerical method from the aspects of stability, dissipation and dispersive errors.

Tightly coupled hyperbolic treatment of buoyant two-phase flow and transport in porous media

Buoyant incompressible multiphase flow and transport in porous media typically is simulated by coupling an elliptic flow equation with a hyperbolic transport description. The tight coupling between flow and transport either requires a fully implicit solution algorithm or very small time steps, if solved sequentially. In any case, however, a large linear system has to be solved each time step.Here a new solution approach is devised, which relies on solving a coupled hyperbolic system of conservation laws with an explicit finite volume method. Consequently, only local operations have to be performed, which is a great advantage in case of massive parallel simulations and if GPUs are employed.The devised method is based on the isothermal Euler equations with momentum source terms accounting for resistance due to the porous medium and buoyancy. In this system the pressure is proportional to the density and in case of very small Mach numbers and relative density variation, the computed velocity field is divergence free and converges towards the mean total volumetric flow in the porous domain. To account for saturation transport, the system was augmented by an additional hyperbolic equation. In order to obtain the numerical fluxes, a characteristic based approximative Riemann solver was developed and 2nd order accuracy in space and time is achieved by piecewise linear reconstruction.Two-phase flow test cases with buoyant plumes and a lock exchange problem demonstrate that the devised hyperbolic approach recovers the correct incompressible solutions. Numerical experiments also demonstrate that the method is accurate and efficient.Thus the devised method is very promising, and conceptually it also can be applied for flow and transport in heterogeneous media; but it remains to be investigated how well it performs for such cases.

PHYSICS-INFORMED NEURAL NETWORKS BASED ON SEQUENTIAL TRAINING FOR CO2 UTILIZATION AND STORAGE IN SUBSURFACE RESERVOIR

CO<sub>2</sub> utilization and storage (CCUS) simulation in subsurface reservoirs with complex heterogeneous structures necessitates a model that can capture multiphase compositional flow and transport. The governing equations are highly nonlinear due to the complex thermodynamic behavior, which involves the appearance and disappearance of multiple phases. Accurate simulation of these processes necessitates the use of stable numerical methods. While machine learning (ML) approaches have been used to solve a variety of nonlinear computational problems, a new approach based on physics-informed neural networks (PINNs) has been proposed for solving partial differential equations (PDEs). Unlike typical ML algorithms that require a large dataset for training, PINNs can train the network with unlabeled data. The applicability of this method has been explored for multiphase flow and transport in porous media. However, for nonlinear hyperbolic transport equations, the solution degrades significantly. This work proposes sequential training PINNs to simulate two-phase transport in porous media. The main concept is to retrain the neural network to solve the PDE over successive time segments rather than train for the entire time domain simultaneously. We observe that sequential training can capture the solution more accurately concerning the standard training for conventional two-phase problems. Furthermore, we extend the sequential training approach for compositional problems in which nonlinearity is more significant due to the complex phase transition. Our approach was tested on miscible and immiscible test cases and showed higher accuracy than the standard training method.

A Novel Sediment Transport Model (STM) Accounting Phase Lag Effect. A Resonance Condition

The classical Exner model coupled with a bed-load sediment flux formula is widely used to describe the morphodynamics of coastal environments. However, the main drawbacks of this model are (i) Lack of robustness, (ii) Lack of differentiation between sediment and fluid velocities, and (iii) Generation of instabilities when the interactions between sediment and fluid flow become more important. Moreover, Exner's model does not allow us to know with which characteristic velocity the bottom is moving. This set of drawbacks weakens the effectiveness of most sediment transport models proposed in the literature, particularly the Exner model. In this work, we reformulate the bed-load equation and we propose a new averaged sediment transport model for application in coastal or estuarine environments. The proposed model incorporates phase shift effects into the bed-load equation. The bedform's characteristic velocity, sediment, and fluid velocity are differentiated. We developed a new first-order, well-balanced, positivity-preserving, path-preserving, and central wind (WBPP-PCCU) scheme to solve the proposed hyperbolic sediment transport model (HSTM). We used the Averaging Essentially Non-Oscillatory (AENO) reconstruction coupled with the third-order Runge-Kutta Semi-Implicit (SI-RK3) method to achieve second-order accuracy. The balance and positivity of the water depth properties were proven. In this work, a resonance condition is proposed. The model facilitates the application of several other schemes such as Roe, HLLC, HLLEM, PVM (polynomial viscosity matrix), RVM (rational viscosity matrix), which require the diagonalization of the Jacobian matrix. The accuracy, robustness, positivity preservation, and equilibrium properties of the resulting model are evaluated using a series of carefully selected test cases. The proposed model provides an excellent ability to simulate sediment transport in a wide range of coastal environments.

Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread

When investigating epidemic dynamics through differential models, the parameters needed to understand the phenomenon and to simulate forecast scenarios require a delicate calibration phase, often made even more challenging by the scarcity and uncertainty of the observed data reported by official sources. In this context, Physics-Informed Neural Networks (PINNs), by embedding the knowledge of the differential model that governs the physical phenomenon in the learning process, can effectively address the inverse and forward problem of data-driven learning and solving the corresponding epidemic problem. In many circumstances, however, the spatial propagation of an infectious disease is characterized by movements of individuals at different scales governed by multiscale partial differential equations. This reflects the heterogeneity of a region or territory in relation to the dynamics within cities and in neighboring zones. In presence of multiple scales, a direct application of PINNs generally leads to poor results due to the multiscale nature of the differential model in the loss function of the neural network. To allow the neural network to operate uniformly with respect to the small scales, it is desirable that the neural network satisfies an Asymptotic-Preservation (AP) property in the learning process. To this end, we consider a new class of AP neural networks for multiscale hyperbolic transport models of epidemic spread that, thanks to an appropriate AP formulation of the loss function, is capable of working uniformly at the different scales of the system. A series of numerical tests for different epidemic scenarios confirms the validity of the proposed approach, highlighting the importance of the AP property in the neural network when dealing with multiscale problems especially in presence of sparse and partially observed systems.

Hydrothermal evaluation of tangent hyperbolic fluid transport and entropy generation through parallel plates

This paper studies the entropy generation and thermal fluid transport of a tangent hyperbolic flow through a vertical microchannel. The fluid mechanics are described with the aid of a coupled system of differential models derived from stress tensors since the hyperbolic tangent fluid exhibits shear thinning behavior. These are analyzed using the Runge-Kutta fourth fifth-order numerical analysis. The result reveals the effect of pertinent parametric on the entropy generated. As shown in the graphical presentation, the impact of heat generation on fluid mass transfer due to viscous dissipation shows the entropy generated is enhanced. Similarly, the entropy is further enhanced along the micro-channel toward the right wall as the convective heat transfer improves. A comparison of results with similar studies in the literature shows a good agreement. The study finds good applications, such as oil recoveries, porous flow in industrial materials, and fluidization beds.

Thermodynamic analysis of a tangent hyperbolic hydromagnetic heat generating fluid in quadratic Boussinesq approximation

The current investigation is to examine the compound impact of electromagnetic induced force and internal heat source on a tangent hyperbolic fluid in quadratic Boussinesq approximation. The current hyperbolic tangent liquid flow and heat transport formulation model adequately predicts and characterizes the shear-stricken event. The nonlinear dimensionless heat transfer flow equations are solved completely using weighted residual solution procedures coupled with Galerkin approximation integration approach. The results in the table and graphs revealed that the magnetic field strength has a substantial impact on the fluid flow and heat propagation, as well as the internal heat source. Therefore, the entropy generation is optimized through an enhanced thermodynamic equilibrium and adequate control of heat generating terms and energy loss.

Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first- and second-order stepping, as well as flows with one, two or three phases.

Output regulation for a first-order hyperbolic PIDE with state and sensor delays

Considering disturbances within domain and at the boundary, a backstepping-based output boundary regulator design is developed for a class of first-order linear hyperbolic partial integro-differential equation (PIDE) in the presence of state and sensor delays. The delays are represented by two transport PDEs, which results in an extended spatial domain where the hyperbolic PIDE, transport PDEs and ordinary differential equation (ODE) are in cascade. The ODE is a finite-dimensional signal model describing exogenous signals. First, a state feedback regulator is realized to achieve a finite time stability by applying an affine Volterra integral transformation. Then, an output regulator is developed on the basis of the nominal plant transfer behavior, which results in an exponential stability. Numerical examples illustrate the performance of the proposed regulators.

VETTAM: a scheme for radiation hydrodynamics with adaptive mesh refinement using the variable Eddington tensor method

ABSTRACT We present Variable Eddington Tensor (VET)-closed Transport on Adaptive Meshes (VETTAM), a new algorithm to solve the equations of radiation hydrodynamics (RHD) with support for adaptive mesh refinement (AMR) in a frequency-integrated, two-moment formulation. The method is based on a non-local VET closure computed with a hybrid characteristics scheme for ray tracing. We use a Godunov method for the hyperbolic transport of radiation with an implicit backwards-Euler temporal update to avoid the explicit time-step constraint imposed by the light-crossing time, and a fixed-point Picard iteration scheme to handle the nonlinear gas-radiation exchange term, with the two implicit update stages jointly iterated to convergence. We also develop a modified wave-speed correction method for AMR, which we find to be crucial for obtaining accurate results in the diffusion regime. We demonstrate the robustness of our scheme with a suite of pure radiation and RHD tests, and show that it successfully captures the streaming, static diffusion, and dynamic diffusion regimes and the spatial transitions between them, casts sharp shadows, and yields accurate results for rates of momentum and energy exchange between radiation and gas. A comparison between different closures for the radiation moment equations, with the Eddington approximation (0th-moment closure) and the M1 approximation (1st-moment closure), demonstrates the advantages of the VET method (2nd-moment closure) over the simpler closure schemes. VETTAM has been coupled to the AMR FLASH (magneto-)hydrodynamics code and we summarize by reporting performance features and bottlenecks of our implementation.

High order modal Discontinuous Galerkin Implicit–Explicit Runge Kutta and Linear Multistep schemes for the Boltzmann model on general polygonal meshes

Deterministic solutions of the Boltzmann equation represent a real challenge due to the enormous computational effort which is required to produce such simulations and often stochastic methods such as Direct Simulation Monte Carlo (DSMC) are used instead due to their lower computational cost. In this work, we show that combining different technologies for the discretization of the velocity space and of the physical space coupled with suitable time integration techniques, it is possible to compute very precise deterministic approximate solutions of the Boltzmann model in different regimes, from extremely rarefied to dense fluids, with CFL conditions only driven by the hyperbolic transport term. To that aim, we develop modal Discontinuous Galerkin (DG) Implicit–Explicit Runge Kutta schemes (DG-IMEX-RK) and Implicit–Explicit Linear Multistep Methods based on Backward-Finite-Differences (DG-IMEX-BDF) for solving the Boltzmann model on multidimensional unstructured meshes. The solution of the Boltzmann collision operator is obtained through fast spectral methods, while the transport term in the governing equations is discretized relying on an explicit shock-capturing DG method on polygonal tessellations in the physical space. A novel class of WENO-type limiters, based on a shifting of the moments of inertia for each zone of the mesh, is used to control spurious oscillations of the DG solution across discontinuities. The use of Linear Multistep Methods (LMM) allows the Boltzmann solutions to be consistent not only with the compressible Euler limit but also with the Navier–Stokes asymptotic regime. In addition, as numerically proven, they also permit to strongly reduce the computational effort compared to Runge–Kutta approaches while maintaining the same or even larger accuracy. The performances of these different time discretization techniques are measured comparing both precision and efficiency. At the same time, comparisons against simpler relaxation type kinetic models such as the BGK model are proposed. The order of convergence is numerically measured for different regimes and found to agree with the theoretical findings. The new methods are validated considering two-dimensional benchmark test cases typically used in the fluid dynamics community. A prototype engineering problem consisting of a supersonic flow around a NACA 0012 airfoil with space–time-dependent boundary conditions is also presented for which the pressure coefficients are measured.

A mathematical model to investigate the effects of fishing zone configurations and mass dependent rates on biomass yield: Application to brown shrimp in Gulf of Mexico

A mathematical model is developed to investigate the effect of harvesting zone configuration upon total biomass yield of a resource species with size dependent continuum spatiotemporal population dynamics. Size (with the interpretation of average mass per individual at each spatiotemporal point) is modeled as an additional state variable along with population density in contrast to the traditional approach of viewing size as a structure parameter. The model takes a form of a coupled system of partial differential equations (PDE) of reaction–diffusion type for the population density and first-order hyperbolic transport type for the mass state variable. The model was investigated both analytically and numerically and is illustrated by applying it to the harvesting of a brown shrimp population in the Gulf of Mexico distributed within a patched network of hypothesized marine protection areas (MPAs) and fishing zones. The results highlight the critical role that size dependent mobility plays in determining optimal MPA network configurations.

Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods

We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic.

A conservative sequential fully implicit method for compositional reservoir simulation

A conservative sequential fully implicit method is derived for compositional reservoir simulation. Multi-phase flow in porous media comprises coupled complex processes: i.e. elliptic flow equation, hyperbolic transport equation and highly nonlinear phase equilibrium equation. These processes contain very different mathematical characteristics that cannot be efficiently solved by one numerical method. As a result, the fully implicit method may become numerically complex and inefficient because the Jacobian includes the derivatives w.r.t. the variables from all of the different processes involved.Jenny et al. (2004) [12] and Lee et al. (2015) [20] demonstrated that flow (pressure) and transport (saturation) for multi-phase flow without compositional effect can be efficiently solved by a sequential fully implicit method. However, the characteristics of the phase equilibrium equations are very different from those of the transport equations. This paper proposes an iterative method that solves the flow, transport and phase equilibrium equations in a sequential manner. The transport of hydrocarbons through porous media is governed by the multi-phase Darcy's equation, which is used to compute the phase velocities. The hydrocarbon components belonging to the same phase are transported with the same phase velocity. Upon arrival in the destination grid cell, these components are redistributed via a phase equilibrium calculation. This observation leads to simplification of the governing equations by reducing primary variables to four (i.e., pressure and three phase saturations). The nonlinear solution scheme composed of the stages outlined above is proven to preserve mass conservation, while a new degree of freedom, “thermodynamic flux”, is introduced to ensure volume conservation. The sequential algorithm is solved iteratively until pressure, saturation, and phase composition are fully converged.It is well-known that sequential solution schemes may require many iterations or fail to converge if the phase equilibrium calculation involves phase transition with a large volume change. This indicates that the current governing equations may not adequately describe fluid flux during rapid phase transition. With numerical examples we demonstrate that such numerical difficulties are successfully resolved via the thermodynamic flux term.

Stabilization of a 2 × 2 system of hyperbolic PDEs with recirculation in the unactuated channel

A result by Guo and Jin (2010) considered boundary control of a string with point damping at the exact midpoint. An interpretation of the model admits a system of first-order hyperbolic (transport) PDEs with actuation at the boundary of one PDE. The unactuated second PDE exhibits recirculation phenomena. The recirculation coupling of these two PDEs gives rise to a stabilization problem with nonlocal terms that prior has not been considered. In this paper, we consider a general class of 2 × 2 hyperbolic PDEs where both the unactuated PDE and the actuated PDE have strict-feedback recirculation (from the outlet in the “upstream direction”). In addition, the state of the unactuated PDE feeds, in a non-local fashion, into the domain of the actuated PDE. We introduce a novel set of transformations through which we arrive at a simple target system with desirable exponential stability characteristics. A backstepping observer and output feedback controller design are also given for this 2 × 2 hyperbolic PDE system. We then apply our control design to the string-and-midway-antidamper application found in Guo and Jin and illustrate the result with simulations.

Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations

In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. The transport equation often has discontinuous solutions and discontinuous inflow boundary conditions, but the normal component of the flux across the mesh interfaces is continuous. In traditional LSFEMs, the continuous finite element space is used to approximate the solution. This will cause unnecessary error around the discontinuity and serious overshooting. In arXiv:1807.01524 [math.NA], we reformulate the equation by introducing a new flux variable to separate the continuity requirements of the flux and the solution. Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, we eliminate the solution from the system and get two flux-only formulations, and develop corresponding LSFEMs. The solution then is recovered by simple post-processing methods using its relation with the flux. These two versions of flux-only LSFEMs use less DOFs than the method we developed in arXiv:1807.01524 [math.NA]. Similar to the LSFEM developed in arXiv:1807.01524 [math.NA], both flux-only LSFEMs can handle discontinuous solutions better than the traditional continuous polynomial approximations. We show the existence, uniqueness, a priori and a posteriori error estimates of the proposed methods. With adaptive mesh refinements driven by the least-squares a posteriori error estimators, the solution can be accurately approximated even when the mesh is not aligned with discontinuity. The overshooting phenomenon is very mild if a piecewise constant reconstruction of the solution is used. Extensive numerical tests are done to show the effectiveness of the methods developed in the paper.

A radiation hydrodynamics scheme on adaptive meshes using the Variable Eddington Tensor (VET) closure

AbstractWe present a new algorithm to solve the equations of radiation hydrodynamics (RHD) in a frequency-integrated, two-moment formulation. Novel features of the algorithm include i) the adoption of a non-local Variable Eddington Tensor (VET) closure for the radiation moment equations, computed with a ray-tracing method, ii) support for adaptive mesh refinement (AMR), iii) use of a time-implicit Godunov method for the hyperbolic transport of radiation, and iv) a fixed-point Picard iteration scheme to accurately handle the stiff nonlinear gas-radiation energy exchange. Tests demonstrate that our scheme works correctly, yields accurate rates of energy and momentum transfer between gas and radiation, and obtains the correct radiation field distribution even in situations where more commonly used – but less accurate – closure relations like the Flux-limited Diffusion and Moment-1 approximations fail. Our scheme presents an important step towards performing RHD simulations with increasing spatial and directional accuracy, effectively improving their predictive capabilities.

Adaptive least-squares finite element methods for linear transport equations based on an H(div) flux reformulation

In this paper, we study the least-squares finite element methods (LSFEM) for the linear hyperbolic transport equations. The linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions, while the normal component of the flux across the mesh faces needs to be continuous. Traditional LSFEMs using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions. In order to separate the continuity requirements, a new flux variable is introduced. With this reformulation, the continuities of the flux and the solution can be handled separately in natural H(div;Ω)×L2(Ω) conforming finite element spaces. Several variants of the methods are developed to handle the inflow boundary condition strongly or weakly.With the reformulation, the new LSFEMs can handle discontinuous solutions and boundary conditions much better than the traditional LSFEMs with continuous polynomial approximations. With least-squares functionals as a posteriori error estimators, the adaptive methods can naturally identify error sources including singularity and non-matching discontinuity. For discontinuity aligned mesh, no extra error is introduced. If an RT0×P0 pair is used to approximate the flux and the solution, the new adaptive LSFEMs can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity.Existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods. Extensive numerical tests are performed to show the effectiveness of the methods developed in the paper.