Non-conservative Equations Research Articles

Manning Coefficient Effect on the Shallow Water Flow Equation

This work will seek to solve shallow water equations, using numerical integration methods, to solve partial differential equations (PDEs) and subsequently present a brief physical interpretation of the solutions. The derivation of the equations is demonstrated, along with the explanation of each physical quantity present. The discretization of non-conservative shallow water equations is carried out using the method of finite differences advanced in time and central in space (FTCS), in explicit form. With the deductions, different Manning coefficients will be addressed in the solutions, verifying their effects.

المحاكاة العددية للتدفقات المضغوطة المكو نة من عنصرين مع المعادلة العامة للحالة

Numerical simulations of compressible two-component flows with a general equation of state (EOS), written in the form of the Mie-Gru¨neisen EOS, have been conducted using the diffuse interface method on a structured mesh. The unsteady and inviscid one-dimensional sixequation model of Kapila is employed to describe compressible two-component flows. The model is hyperbolic and non-conservative. The solution of the hyperbolic equations, including the non-conservative equation for the volume fraction evolution, is obtained using an extended Harten-Lax-Leer (HLL) approximate Riemann solver. A general formulation for various EOSs is proposed to enable simulations of a wide range of applications. In which the two constituents are either governed by the same EOS or different types of EOS. In this model, both fluids have the same velocity, but each fluid has its own pressure. Therefore, the pressure relaxation process is performed instantaneously to drive both constituents' pressures towards equilibrium. Several computational test problems were conducted in one and two dimensions to show the ability of the numerical method to simulate such problems. The computed results are presented without introducing spurious pressure oscillations in the method.

Treatment for the Singularity Problem in the Phase Segregation

In the Eulerian–Eulerian two-fluid model, comprising air and droplets, singularities often arise as the phase fraction of dispersed droplets approaches zero. The nonconservative momentum equations can address this issue, but it will lead to inaccurate solutions when encountering discontinuities. To tackle the singularity problem and ensure accurate solutions for the droplet field, a novel adaptive form of momentum equations is proposed. The adaptive form defaults to the conservative form and smoothly transitions to the phase-intensive form when the phase fraction falls below a critical value, specifically 1/1000 of the inlet. The LU-SGS algorithm, enhanced with the inner iteration step method, is employed for numerical solutions of the droplet field. Comparative analysis reveals that the discrepancy between the results obtained from the adaptive form and the conservative form is negligible, under 0.5‰. Notably, with larger Courant numbers, the conservative form diverges from waterless regions despite imposing the minimum phase fraction constraint. Conversely, the adaptive form consistently maintains a Courant number of five times or more, potentially exceeding a hundredfold in complex flows. Although the adaptive form moderately increases complexity and computational time for a single time step, the improvement of the maximum Courant number will bring significant computational efficiency advantages. Moreover, the adaptive form holds promise for application in various other multiphase flow scenarios.

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given.

Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

TUMME 2023: Tsinghua University Minnesota Master Equation program. New version announcement

This is an updated version of TUMME (Tsinghua University Minnesota Master Equation program), which is a computer program for setting up and solving master equations for chemical kinetics of unimolecular and bimolecular reactions. The master equation is discretized in terms of energy bins, and rate constants are calculated from chemically significant eigenmodes. TUMME has interfaces to Gaussian 16, Polyrate 2023, and MSTor 2023 output files that allow the master equation code to obtain the microcanonical flux coefficients needed for the kernel of the master equation as calculated by conventional transition state theory (TST), variational transition state theory (VTST) with various tunneling methods, or multi-structural or multi-path VTST (MS-VTST or MP-VTST) with various tunneling methods. The tunneling methods supported include zero-curvature tunneling (ZCT), small-curvature tunneling (SCT), large-curvature tunneling (LCT), and microcanonically optimized multidimensional tunneling (μOMT). For mechanisms involving only unimolecular isomerization (no bimolecular pairs), TUMME can set up and solve a conservative master equation for both rate constants and time-dependent energy-bin populations. For mechanisms involving bimolecular pairs, TUMME 2023 can set up and solve two kinds of master equation: (i) a nonconservative master equation for calculating rate constants of bimolecular reactions and (ii) a conservative master equation that includes bimolecular association in the transition matrix and that can be used for calculating the time evolution of the concentration of a pseudo-first-order bimolecular reactant. TUMME is written in double precision with Python 3; quadruple and octuple precision are also available for some subtasks in C++. The Python code can run in serial or parallel (MP or MPI), and C++ code can run on a single processor or on multiple processors with OpenMP. The program includes a manual and a tutorial. New version program summaryProgram Title: TUMME 2023CPC Library link to program files:https://doi.org/10.17632/whcnvm2mc9.2Developer's repository link:https://doi.org/10.5281/zenodo.7943283Licensing Provisions: Apache-2.0 for the program and CC-BY-4.0 for the manualProgramming languages: Python 3 and C++Does the new version supersede the previous version?: YesJournal reference of previous version: Comput. Phys. Commun. 270 (2022) 108140 [1]External libraries: Numpy, Scipy, Numba, mpi4py (optional), modified mpack (optional), qd (optional), omp (optional)Interfaces to other programs: Gaussian 16 [2], Polyrate 2023 [3], and/or MSTor 2023 [4]Nature of problem: Characterize and calculate rate constants for a temperature-dependent and pressure-dependent complex reaction systemSolution method: Solve the energy-grained master equation based on the chemically significant eigenmodes to get phenomenological rate constants and time evolutions of the populations [1,5–7].Reason for new version: The new version increases the functionality and accuracy, and it offers an easier way to run the program. In addition, some bugs have been corrected.Summary of revisions:1.Added an option (available in high precision) to treat pseudo-first-order bimolecular reactions to obtain the time evolution of the reactants and intermediates. [8]2.Added an inverse-Laplace-transform option in the REACTION block.3.Abandoned rounding the sizes of energy bins to an integer wavenumber (ESOT).4.Changed the way to run TUMME by providing an executable bash script.5.Slightly adjusted the strategy for eigenmode mergers to deal with some unexpected cases.6.Adjusted the output for the partition functions.7.Added options for large-curvature tunneling (LCT) and microcanonically optimized multidimensional tunneling (μOMT).8.Added the keyword pyrfile to support a Polyrate 2023 [3] file that ends with a suffix .fu100.9.Fixed some bugs.10.Updated the manual and added a tutorial to the program distribution, so the distribution now has both a manual and a tutorial.

Gradient-descent-like scheme for the Allen–Cahn equation

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes.

Stability analysis of axial geometry with anisotropic background in f(R,T) gravity

In this paper, we highlight the variables preserving stability of a very restricted class of anisotropic axial symmetrical compact geometry in the scenario of [Formula: see text] gravity, where [Formula: see text] stands for energy–momentum tensor’s trace and [Formula: see text] is invariant Ricci curvature. In the framework of [Formula: see text] gravity, we set up field equations as well as non-conservation equations. We use a perturbation technique for all variables involved in non-conservation equations, field equations, extra curvature terms of modified gravity as well as for considered gravity model (i.e. [Formula: see text]) to evaluate the collapse equation. We establish certain significant constraints for the stiffness parameter [Formula: see text] in Newtonian [Formula: see text] and post-Newtonian [Formula: see text] approximation to study the dynamical instability of a stellar compact configuration. In order to preserve the stability of an anisotropic self-gravitating axially symmetric configuration, we place certain restrictions on physical quantities. To examine the stable and unstable behavior of considered geometry via graphical approaches, we include schematic diagrams at the [Formula: see text] and [Formula: see text] eras.

A New Family of Thermodynamically Compatible Discontinuous Galerkin Methods for Continuum Mechanics and Turbulent Shallow Water Flows

In this work we propose a new family of high order accurate semi-discrete discontinuous Galerkin (DG) finite element schemes for the thermodynamically compatible discretization of overdetermined first order hyperbolic systems. In particular, we consider a first order hyperbolic model of turbulent shallow water flows, as well as the unified first order hyperbolic Godunov–Peshkov–Romenski (GPR) model of continuum mechanics, which is able to describe at the same time viscous fluids and nonlinear elastic solids at large deformations. Both PDE systems treated in this paper belong to the class of hyperbolic and thermodynamically compatible systems, since both satisfy an entropy inequality and the total energy conservation can be obtained as a direct consequence of all other governing equations via suitable linear combination with the corresponding thermodynamic dual variables. In this paper, we mimic this process for the first time also at the semi-discrete level at the aid of high order discontinuous Galerkin finite element schemes. For the GPR model we directly discretize the entropy inequality and obtain total energy conservation as a consequence of a suitable discretization of all other evolution equations. For turbulent shallow water flows we directly discretize the nonconservative evolution equations related to the Reynolds stress tensor and obtain total energy conservation again as a result of the thermodynamically compatible discretization. As a consequence, for continuum mechanics the new DG schemes satisfy a cell entropy inequality directly by construction and thanks to the discrete thermodynamic compatibility they are provably nonlinearly stable in the energy norm for both systems under consideration.

Semi-implicit methods for advection equations with explicit forms of numerical solution

We present a parametric family of semi-implicit second order accurate numerical methods for non-conservative and conservative advection equation for which the numerical solutions can be obtained in a fixed number of forward and backward alternating substitutions. The methods use a novel combination of implicit and explicit time discretizations for one-dimensional case and the Strang splitting method in several dimensional case. The methods are described for advection equations with a continuous variable velocity that can change its sign inside of computational domain. The methods are unconditionally stable in the non-conservative case for variable velocity and for variable numerical parameter. Several numerical experiments confirm the advantages of presented methods including an involvement of differential programming to find optimized values of the variable numerical parameter.

Entropy-conserving Scheme for Modeling Nonthermal Energies in Fluid Dynamics Simulations

We compare the performance of energy-based and entropy-conserving schemes for modeling nonthermal energy components, such as unresolved turbulence and cosmic rays, using idealized fluid dynamics tests and isolated galaxy simulations. While both methods are aimed to model advection and adiabatic compression or expansion of different energy components, the energy-based scheme numerically solves the nonconservative equation for the energy density evolution, while the entropy-conserving scheme uses a conservative equation for modified entropy. Using the standard shock tube and Zel’dovich pancake tests, we show that the energy-based scheme results in a spurious generation of nonthermal energy on shocks, while the entropy-conserving method evolves the energy adiabatically to machine precision. We also show that, in simulations of an isolated L ⋆ galaxy, switching between the schemes results in ≈20%–30% changes of the total star formation rate and a significant difference in morphology, particularly near the galaxy center. We also outline and test a simple method that can be used in conjunction with the entropy-conserving scheme to model the injection of nonthermal energies on shocks. Finally, we discuss how the entropy-conserving scheme can be used to capture the kinetic energy dissipated by numerical viscosity into the subgrid turbulent energy implicitly, without explicit source terms that require calibration and can be rather uncertain. Our results indicate that the entropy-conserving scheme is the preferred choice for modeling nonthermal energy components, a conclusion that is equally relevant for Eulerian and moving-mesh fluid dynamics codes.

Generalized nonconservative gravitational field equations from Herglotz action principle

We formulate a nonconservative gravitational theory based on the Herglotz variational principle in a tensorial covariant form. The model presented here may be seen as an improvement of the theory proposed in [ et al., Phys. Rev. D 95, 101501 (2017)], whose resulting field equations are meaningful just in particular coordinate systems. The new theory we report in this work is free from such a restriction. In comparison to the standard general relativity, both theories based on the Herglotz principle contain an extra vector field, the Herglotz field, but the new theory is formulated by taking advantage of the restricted equivalence between Lagrangian functions in the scope of the Herglotz action principle. The more restricted class of equivalent Lagrangian functions, in comparison with the Hamilton variational principle, is the key point to finding a Lagrangian that generates a new alternative gravitational theory in a covariant form. Once the equations that govern the dynamics of the gravitational field are obtained, a few simple cosmological models are investigated. It is found that the Herglotz vector field reduces to a single function that, under certain conditions, plays the role of the cosmological constant in general relativity, turning unnecessary the introduction of dark energy to explain the accelerated expansion of the Universe. The linearized version of the theory is also investigated and it is verified that the theory shows a dissipative character regarding the propagation of gravitational perturbations. From observational data, in both scenarios, the magnitude of the Herglotz field is estimated.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Numerical Calculations of the GRP Scheme for Nonconservative Ideal Fluid Mechanics Equations Relying on Parallel Calculation of Multifluid Grids

In order to solve the numerical method of nonconservative ideal hydrodynamics equations, the viscous perturbation technique for solving nonconservative hydrodynamics equations is improved and tested by solving the Riemann problem. The calculation of nonconservative ideal fluid mechanics is based on the GRP format. This article aims at the calculation method of nonconservative ideal fluid mechanics in the GRP format. Riemann and the corresponding periodic vortex are processed. The multifluid network processing method in the article is compared with the current method. The result can prove that this format can be used to solve the nonconservative ideal fluid dynamics equation of multiple values in the GRP format group, its computing power is strong, and the result of the solution is accurate.

A splitting scheme for the coupled Saint-Venant-Exner model

We present a splitting method for the one-dimensional Saint-Venant-Exner equations used for describing the bed evolution in shallow water systems. We adapt the flux vector splitting approach of Toro and Vázquez-Cendón (2012) and identify one subsystem of conservative equations (advection system) and one of non-conservative equations (pressure system), both having a very simple eigenstructure compared to the full system. The final numerical scheme is constructed using a Godunov-type path-conservative scheme for the pressure system and a simple conservative Godunov method for the advection system and solved following a coupled solution strategy. The resulting first-order accurate method is extended to second order of accuracy in space and time via the ADER approach together with an AENO reconstruction technique. Accuracy, robustness and well-balanced properties of the resulting scheme are assessed through a carefully selected suite of testcases. The scheme is exceedingly simple, accurate and robust as the sophisticated Godunov methods. A distinctive feature of the novel scheme is its flexibility in the choice of the sediment transport closure formula, which makes it particularly attractive for scientific and engineering applications.

High Order Finite Volume Schemes for Solving the Non-Conservative Convection Equations on the Unstructured Grids

High Order Finite Volume Schemes for Solving the Non-Conservative Convection Equations on the Unstructured Grids

Non-conservative effects on spinning black holes from world-line effective field theory

We generalize the worldline EFT formalism developed in [4–9] to calculate the non-conservative tidal effects on spinning black holes in a long wavelength approximation that is valid to all orders in the magnitude of the spin. We present results for the rate of change of mass and angular momentum in a background field and find agreement with previous calculations obtained by different techniques. We also present new results for both the non-conservative equations of motion and power loss/gain for a binary inspiral, which start at 5PN and 2.5PN order respectively and manifest the Penrose process.

A MOOD-MUSCL Hybrid Formulation for the Non-conservative Shallow-Water System

The stability of the high-order finite volume method for hyperbolic systems is based on non-linear procedures that prevent the creation of non-physical oscillations. Traditional techniques use the a priori paradigm where the procedure is carried out with the current time step solution. The a posteriori paradigm lies on an advanced in time candidate solution and a posterior evaluation of its stability, followed by a cure when necessary. To compare the two strategies, we propose a detailed study using the non-conservative shallow-water equations as a prototype, where both techniques are applied together in the framework of second-order linear reconstruction for the sake of simplicity. Then, a hybrid version combining the most positive aspects of both methods is proposed and analysed.

Qualitative Properties of the Collisional Model for Describing Shock-Wave Dynamics in Gas Suspensions

A The paper presents theoretical analysis of a two-phase medium model for describing shock-wave processes in dense gas particle mixtures, taking into account the chaotic motion and collisions of particles. Hyperbolicity regions and composite-type domains are identified in the governing system of equations. It is shown that the hyperbolicity region expands beyond the collisionless model. An approximate hyperbolized model is presented. Numerical solutions of the problem on the development of various shock-wave structure types are compared. The convergence properties in numerical simulations of non-conservative composite-type equations were established using the Harten and Gentry–Martin–Daly schemes. The conditions of the hyperbolized model applicability to different types of flows are obtained. It is shown that, in the general case, shock-wave processes in gas suspensions should preferably be analyzed within a complete model.

Simulation of Melt Viscosity Effect on the Rate of Solidification in Polymer

Phase field model has been successfully derived from ordinary metal phase field equation to simulate the behavior of semi-crystalline polymer solidification phenomenon. To obtain the polymer phase field model, a non-conserved phase field equation can be expanded to include the unique polymer parameters, which do not exist in metals, for example, polymer melt viscosity and diffusion coefficient. In order to expand this model, we include free energy density and non-local free energy density based on Harrowel-Oxtoby and Ginzburg-Landau theorem for polymers. The expansion principle for a higher order of binary phase field parameter was employed to obtain fully modified phase field equation. To optimize the final properties of the products, the solidification phenomenon in polymers is very important. Here, we use our modified equation to investigate the effect of melt viscosity on the rate of solidification by employing ordinary differential equation numerical methods. It was found that the rate of solidification is related to the melting temperature and the kinetic coefficient.