Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

The generalized quasi-variational principles of non-conservative systems with two kinds of variables

According to the corresponding relations between general forces and general displacements, the balancing and geometrical equations of elasticity are multiplied by the corresponding virtual quantities, integrated with volume and area, and then added algebraically. Proceeding to the next step, by substituting constitutive relation and considering that body force and surface force are both fellow forces, the generalized quasi-variational principles with the two kinds of variables of the first type are established in non-conservative systems. Through substituting another constitutive relation, using similar methods as above, the generalized quasi-variational principles with the two kinds of variables of the second type are established in non-conservative systems. By using the generalized quasi-complementary energy principles with the two kinds of variables of the first type, a method for solving two kinds of variables (internal force and deformation) is given for non-conservative systems of the typical fellow forces.

Noether's theorem for non-conservative Hamilton system based on El-Nabulsi dynamical model extended by periodic laws

This paper focuses on the Noether symmetries and the conserved quantities for both holonomic and nonholonomic systems based on a new non-conservative dynamical model introduced by E1-Nabulsi. First, the E1-Nabulsi dynamical model which is based on a fractional integral extended by periodic laws is introduced, and E1-Nabulsi-Hamilton＇s canoni- cal equations for non-conservative Hamilton system with holonomic or nonholonomic constraints are established. Second, the definitions and criteria of E1-Nabulsi-Noether symmetrical transformations and quasi-symmetrical transformations are presented in terms of the invariance of E1-Nabulsi-Hamilton action under the infinitesimal transformations of the group. Fi- nally, Noether＇s theorems for the non-conservative Hamilton system under the E1-Nabulsi dynamical system are established, which reveal the relationship between the Noether symmetry and the conserved quantity of the system.

Conformal invariance and conserved quantities of non-conservative Lagrange systems by point transformations

This paper studies the conformal invariance by infinitesimal point transformations of non-conservative Lagrange systems. It gives the necessary and sufficient conditions of conformal invariance by the action of infinitesimal point transformations being Lie symmetric simultaneously. Then the Noether conserved quantities of conformal invariance are obtained. Finally an illustrative example is given to verify the results.

A new type of adiabatic invariants for disturbed non-conservative nonholonomic system

According to the Herglotz variational principle and differential variational principle of Herglotz type, we study the adiabatic invariants for a non-conservative nonholonomic system. Firstly, the differential equations of motion of the non-conservative nonholonomic system based upon the generalized variational principle of Herglotz type are given, and the exact invariant for the non-conservative nonholonomic system is introduced. Secondly, a new type of adiabatic invariant for the system under the action of a small perturbation is obtained. Thirdly, the inverse theorem of the adiabatic invariant is given. Finally, an example is given.

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non- conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invaxiance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.

Question of Parity Non-conservation in Weak Interactions

In order to reasonably explain the phenomenon of cell bioelectricity,we proposed the conservation law of cell membrane area,established the ion inequality equation,and therefore paid attention to the mystery of“θ-τ”.We researched and analyzed the“θ-τ”mystery,discussed the parity non-conservation in weak interactions,suggested possible experiments to test the parity non-conservation in weak interactions,and gave our research and analysis conclusions:The parity non-conservation in weak interactions,is still a“conjecture”;The experimental scheme suggested in the papers by C.N.Yang et al.cannot determine whether the weak interaction can separate left and right,and it is impossible to directly answer whetherθandτin the“θ-τ”mystery are the same particle;The Co60βdecay experiment such as C.S.Wu is a pseudo-mirror experiment,whether the experimental result violates parity conservation is only based on the assumption of C.N.Yang et al.In fact,experiments such as polarized Co60 did not overturn the so-called“law of parity conservation”.The mirror image principle does not have any physical meaning,does not correspond to any physical conservation quantity,and cannot be destroyed by any physical experiment.In the process of turning“mirror symmetry”and“mirror asymmetry”into so-called physical“common sense”and scientific“facts”respectively,the methods of transformation are“stealing concepts”and“circular argumentation”.The“θ-τ”mystery is a“man-made”mystery.θandτare two different particles,which may be the result of the same precursor particle being divided into two.The work of C.N.Yang,T.D.Lee,C.S.Wu et al.has brought quantum physicists from the“small black room”to the“bigger black room”or“smaller black room”.The right and wise choice is to go back through the door that came in.With the development of science today,it is time for some contents to reform from the bottom.

Holographic dark energy in non-conserved gravity theory

Recently, reconsidering the Rastall idea T_(μ;νν)^(v)=α_(,μ) through relativistic thermodynamics gives a new form for the scalar field α which led us to construct modern modified theory of gravity debugged ‘non-conserved gravity theory’ Fazlollahi 2023 Euro. Phys. J. C 83 923. This theory unlike other modified theories of gravity cannot directly explain the current acceleration expansion in the absence of the cosmological constant and or existence of other forms of dark energy. Hence, in this study we have reinvestigated holographic dark energy Ρ_(X)~H^(2) in the non-conserved theory of gravity. In this context, the density and pressure of dark energy depend on the non-conserved term and density of the dust matter field. As shown, due to non-conservation effects on large-scale structures, unlike the original holographic model, our model onsets an acceleration epoch for the current Universe satisfies observations. Moreover, the interaction and viscous scenarios are studied for this model.

Non-linear and non-conservative quasi-variational principle of flexible body dynamics and application in spacecraft dynamics

The law of conservation of energy is one of the most fundamental laws of nature.According to the law of the conservation of energy,the non-linear and non-conservative quasi-variational principle of flexible body dynamics is established.The physical meaning of the quasi-stationary value conditions has been explained in non-linear and non-conservative flexible body dynamics.In the case study,the application in spacecraft dynamics is researched.

Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping

In this paper,the symplectic perturbation series methodology of the non-conservative linear Hamiltonian system is presented for the structural dynamic response with damping.Firstly,the linear Hamiltonian system is briefly introduced and its conservation law is proved based on the properties of the exterior products.Then the symplectic perturbation series methodology is proposed to deal with the non-conservative linear Hamiltonian system and its conservation law is further proved.The structural dynamic response problem with eternal load and damping is transformed as the non-conservative linear Hamiltonian system and the symplectic difference schemes for the non-conservative linear Hamiltonian system are established.The applicability and validity of the proposed method are demonstrated by three engineering examples.The results demonstrate that the presented methodology is better than the traditional Runge–Kutta algorithm in the prediction of long-time structural dynamic response under the same time step.

High Order Accurate Direct Arbitrary-Lagrangian-Eulerian ADER-MOOD Finite Volume Schemes for Non-Conservative Hyperbolic Systems with Stiff Source Terms

In this paper we present a 2D/3D high order accurate finite volume scheme in the context of direct Arbitrary-Lagrangian-Eulerian algorithms for general hyperbolic systems of partial differential equations with non-conservative products and stiff source terms.This scheme is constructed with a single stencil polynomial reconstruction operator,a one-step space-time ADER integration which is suitably designed for dealing even with stiff sources,a nodal solver with relaxation to determine the mesh motion,a path-conservative integration technique for the treatment of non-conservative products and an a posteriori stabilization procedure derived from the so-called Multidimensional Optimal Order Detection(MOOD)paradigm.In this work we consider the seven equation Baer-Nunziato model of compressible multi-phase flows as a representative model involving non-conservative products as well as relaxation source terms which are allowed to become stiff.The new scheme is validated against a set of test cases on 2D/3D unstructured moving meshes on parallel machines and the high order of accuracy achieved by the method is demonstrated by performing a numerical convergence study.Classical Riemann problems and explosion problems with exact solutions are simulated in 2D and 3D.The overall numerical code is also profiled to provide an estimate of the computational cost required by each component of the whole algorithm.

POSTBUCKLING RESPONSE OF CANTILEVER BEAMS OF NON-LINEAR ELASTIC MATERIAL SUBJECTED TO SUBTANGENTIAL FOLLOWER FORCES

This paper deals with the problem of the postbuckling response of a thin cantilever beam ofnon-linear material, subjected to subtangential follower forces. Based on the well-knownBernoulli-Euler bending moment-curvature relation, the proposed problem is reduced to a specialeigenvalue problem of non-linear differential equation. An approximate solution is achieved byusing a simple and very effective technique, which leads to reliable results even in the case of verylarge deflections. The initial postbuckling path depending on the subtangential follower forces inequilibrium is then obtained. Moreover, the individual and coupling effect of the subtangential fol-lower force, the material non-linearity and the beam slenderness ratio on the initial postbucklingpath are also discussed in detail.

Power flows of three non-conservatively series-coupled oscillators, Part Ⅱ: Numerical analysis

The direct and indirect power flows among three non-conservatively series-coupled oscillators subjected to random forces have been investigated by the numerical method. The relationship between the power flows and the parameters of the oscillators and couplings have been studied. It is shown that the indirect power flow is caused by the resonant transmission between the indirectly coupled oscillators. Once the parameters of the oscillator elements have been settled, the indirect power flow is mainly determined by the coupling stiffness k3 and k4, and is less infiuenced by coupling dampings c3 and c4. The indirect power flow cannot be ignored in strong coupling conditions. Because of the resonant transmission between the indirectly coupled oscillators, the indirect power flow still cannot be neglected even if the coupling stiffness is relatively small and the coupling damping is relatively great

The coupling loss factors for non-conservatively coupled structures

The method to calculate the coupling loss factors for non-conservatively coupled structures by using the mobilities of substructures is studied. By using the form of the energy balance equations of conservatively coupled systems, the relationship between the coupling loss factors and the energy ratios in non-conservatively coupled systems is derived. The method to calculate the energy ratios by using the mobilities of substructures is introduced, and experiment verification is carried out. The test data agree well with the predicted results.

Thermodynamic and rate variational formulation of models for inhomogeneous gradient materials with microstructure and application to phase field modeling

In this work,thermodynamic models for the energetics and kinetics of inhomogeneous gradient materials with microstructure are formulated in the context of continuum thermodynamics and material theory.For simplicity,attention is restricted to isothermal conditions.The materials of interest here are characterized by（1） first- and secondorder gradients of the deformation field and（2） a kinematic microstructure field and its gradient（e.g.,in the sense of director,micromorphic or Cosserat microstructure）.Material inhomogeneity takes the form of multiple phases and chemical constituents,modeled here with the help of corresponding phase fields.Invariance requirements together with the dissipation principle result in the reduced model field and constitutive relations.Special cases of these include the wellknown Cahn-Hilliard and Ginzburg-Landau relations.In the last part of the work,initial boundary value problems for this class of materials are formulated with the help of rate variational methods.

Observation of nonconservation characteristics of radio frequency noise mechanism of 40-nm n-MOSFET

Bias non-conservation characteristics of radio-frequency noise mechanism of 40-nm n-MOSFET are observed by modeling and measuring its drain current noise. A compact model for the drain current noise of 40-nm MOSFET is proposed through the noise analysis. This model fully describes three kinds of main physical sources that determine the noise mechanism of 40-nm MOSFET, i.e., intrinsic drain current noise, thermal noise induced by the gate parasitic resistance, and coupling thermal noise induced by substrate parasitic effect. The accuracy of the proposed model is verified by noise measurements, and the intrinsic drain current noise is proved to be the suppressed shot noise, and with the decrease of the gate voltage, the suppressed degree gradually decreases until it vanishes. The most important findings of the bias non-conservative nature of noise mechanism of 40-nm n-MOSFET are as follows.（i） In the strong inversion region, the suppressed shot noise is weakly affected by the thermal noise of gate parasitic resistance. Therefore, one can empirically model the channel excess noise as being like the suppressed shot noise.（ii） In the middle inversion region, it is almost full of shot noise.（iii） In the weak inversion region, the thermal noise is strongly frequency-dependent, which is almost controlled by the capacitive coupling of substrate parasitic resistance. Measurement results over a wide temperature range demonstrate that the thermal noise of 40-nm n-MOSFET exists in a region from the weak to strong inversion, contrary to the predictions of suppressed shot noise model only suitable for the strong inversion and middle inversion region. These new findings of the noise mechanism of 40-nm n-MOSFET are very beneficial for its applications in ultra low-voltage and low-power RF, such as novel device electronic structure optimization, integrated circuit design and process technology evaluation.

Principle of statistical energy analysis for non-conservatively coupled systems

Traditional Statistical Energy Analysis (SEA) theory can not deal with dynamic problems concerned with non-conservatively coupled systems. In this paper, based on the theory of power flow between them and energy distribution in non-conservatively coupled osillators, equations of power balance and those for calculation of each concerned power flow and other power items are derived to develop SEA theory for non-conscrvativcly coupled systems. Results show that conservative coupling is only a special case of non-conservative coupling situations, effect of coupling damping on power flow and energy distribution in non-conservatively coupled systems arc not negligible unless coupling damping is much smaller compared with internal one. As an application of the theory, energy problems of non-conservatively coupled plates are studied theoretically and experimentally.

Power flows of three non-conservatively series-coupled oscillators,Part Ⅰ:Theory

The formulae of power flows among three non-conservatively series-coupled oscillators have been derived. It is shown that the power flows among the three non-conservatively series-coupled oscillators, similar to that of the three conservatively series-coupled oscillators,include the direct and indirect power flows. The direct and indirect power flows are proportional to the differences of the stored energies in the directly coupled and indirectly coupled oscillators. The proportionality constants are functions of parameters of the oscillators and couplings

Problems of Nonlinear Computational Instability in Evolution Equations

Some problems of nonlinear computational instability are discussed in this article, which are shown as follows: 1) Three types of representative evolution equations are analyzed, and the close relationship between the nonlinear computational stability or instability in their corresponding difference equations and the properties of their solution is revealed. 2) The problem of nonlinear computational instability in conservative differencing equations with the periodic boundary condition is further discussed, and some effective ways to avoid nonlinear computational instability are proposed. 3) The problem of nonlinear computational instability in non-conservative difference equations with the aperiodic boundary condition is focused on by using nonlinear advection equations as examples, and u synthetic analysis method' is given to judge their computational stability.

Weak- and hyperfine-interaction-induced 1s2s ~1S_0→ 1s^2 ~1S_0 E1 transition rates of He-like ions

Weak- and hyperfine-interaction-induced 1 s2s 1S0→ 1S2 1 S0 E 1 transition rates for the isoelectronic sequence of Helike ions have been calculated using the multi-configuration Dirac-Hartree-Fock （MCDHF） and relativistic configuration interaction methods. The results should be helpful for the future experimental investigations of parity non-conservation effects.