Vector Field Method Research Articles

The hidden null structure of the compressible Euler equations and a prelude to applications

We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to transport equations for the Cartesian components of the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the “wave part” of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify the strong null condition. The latter is a nonlinear condition signifying the complete absence of nonlinear interactions involving more than one differentiation in a direction transversal to the acoustic characteristics. Moreover, the same good structures are found in the equations verified by the Euclidean divergence and curl of the specific vorticity. This is important because one needs to combine estimates for the divergence and curl with elliptic estimates to obtain sufficient regularity for the specific vorticity, whose derivatives appear as inhomogeneous terms in the wave equations. The structures described above collectively open the door for our companion results, in which we exhibit a stable regime of initially smooth solutions that develop a shock singularity. In particular, the first Cartesian coordinate partial derivatives of the velocity and density blow up, while relative to a system of geometric coordinates adapted to the acoustic characteristics, the solution (including the vorticity) remains many times differentiable, all the way up to the shock. The good null structures, which are often associated with global solutions, are in fact key to proving that the shock singularity forms. Our secondary goal in this paper is to provide an overview of the central role that the structures play in the proof.

Decay of Maxwell fields on Reissner–Nordström–de Sitter black holes

In this paper we use Morawetz estimates with geometric energy estimates -the so-called vector field method- to prove decay results for the Maxwell field in the static exterior region of the Reissner-Nordstr{\o}m-de Sitter black hole. We prove two types of decay: The first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of $t^{-1}$ which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field. The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper.

A vector field method for radiating black hole spacetimes

We develop a commuting vector field method for a general class of radiating spacetimes. The metrics considered are certain long range perturbations of Minkowski space including those constructed from global stability problems in general relativity. Our method provides sharp peeling estimates for solutions to both linear and nonlinear (null form) scalar fields.

Hierarchical Collaboration Approach for Tracking of Flying Target by UAVs

This paper investigates a hierarchical target tracking method based on the collaboration of unmanned aerial vehicles (UAVs) at different altitudes. To enhance the target tracking range, the high-altitude UAVs monitor the wide area, and transmit their surveillance information to low-altitude UAVs which directly detect and collect information about the target's movements. The contributions of this paper are threefold: First, to track the flying target in a dynamic environment, a modified Lyapunov guidance vector field (LGVF) method is used to plan velocities for UAVs, where a time-varying vertical component is incorporated into the traditional LGVF function to satisfy the constraints of cluster communication. Secondly, a three-dimensional local collision-free guidance vector field (TLCGVF) method is proposed for UAVs to plan collision-free paths on line. To simultaneously track the target and avoid obstacles, the vector field by LGVF is used as the original vector field of TLCGVF. Thirdly, the rolling optimization strategy is used to adjust the reactive parameters of TLCGVF to enhance the path quality. The simulation results confirm the feasibility of the above approach.

Obstacle Avoidance Model for UAVs with Joint Target based on Multi-Strategies and Follow-up Vector Field

A multi-strategies obstacle avoidance method based on follow-up rotating vector field is proposed for UAV flight planning. The inter-UAVs repulsion with distance factor and the target directional gravity are used as the inter-UAVs control strategy. In the process of obstacle avoidance, the artificial potential field method is used outside the obstacle to guide the UAVs to sail to the target point. In the obstacle avoidance scope, the follow-up rotating vector field method is proposed to avoid convex polyhedral obstacles. Finally, smoothing strategy is conbined to the follow-up rotating vector field to obtain safety and smoothing path. The simulation results show that the proposed algorithm can solve the problem of “dead zone” and “jitter” in the track by combining various strategies, and realize the obstacle avoidance behavior while avoiding collision between machines.

A vector field method for massless relativistic transport equations and applications

In this article, we present a vector field method for the study of solutions to massless relativistic transport equations. Compared to [10], we remove the Lorentz boosts of the commutation vector fields and we prove a new functional inequality in order to derive pointwise decay estimates on the solutions. It makes our method more suitable for the study of systems coupling a massless Vlasov equation with an equation which has a different speed of propagation. We also believe that our approach can be adapted more easily to the context of curved backgrounds.In the second part of this paper, we apply our method in order to derive sharp decay estimates on the spherically symmetric small data solutions of the relativistic massless Vlasov-Poisson system. No compact support assumption is required on the data and, in particular, the initial decay in the velocity variable is optimal. In order to compensate the slow decay rate of the particle density near the light cone, we take advantage of the null structure of the equations.

Novel high-order energy-preserving diagonally implicit Runge–Kutta schemes for nonlinear Hamiltonian ODEs

We utilize the invariant energy quadratization approach to transform the nonlinear Hamiltonian ODE into an equivalent form which has a quadratic energy. Then the reformulation is discretized using a class of diagonally implicit Runge–Kutta schemes. The proposed schemes are proved to conserve a modified quadratic energy conservation law and converge with high-order accuracy. Numerical experiments for several ODEs are provided to illustrate the accuracy, convergence and conservative properties of the proposed method. Comparisons with the symplectic Runge–Kutta scheme and energy-preserving average vector field method are presented to demonstrate the energy-preserving property and accuracy of the proposed method.

A Lateral-Directional Control Method for High Aspect Ratio Full-Wing UAV and Flight Tests

To solve the lateral-directional control problem of the high aspect ratio full-wing unmanned aerial vehicle (UAV) without an aileron and rudder, a control method is proposed that uses the differential thrust of propellers as the control output and the yaw angle as the controlled attitude angle. Meanwhile, simulation analysis and experimental verification are carried out. First, a lateral-direction mathematical model and a differential thrust of propeller model of the full-wing drone are established. The influence of the aerodynamic derivative C Y β on the lateral-direction mode is analyzed. Second, based on nonlinear dynamic inversion (NDI) and active disturbance rejection control (ADRC) theories, a yaw angle controller that uses the differential thrust of propellers as the control output is designed. Finally, the vector field (VF) method is improved to obtain the straight-line trajectory tracking method satisfying different speeds, and the logic of waypoint switching is given. The research shows that C Y β has a great influence on the dutch roll damping of the drone. For the full-wing configuration, it is feasible to use the yaw angle as the controlled attitude angle without considering the roll angle. The simulation and experimental results show that the designed lateral-directional control method for the high aspect ratio full-wing UAV has a good control effect and disturbance rejection ability. Meanwhile, the control method has less parameters to adjust and less calculation, which is very suitable for engineering applications.

Singular Perturbed Vector Field (SPVF) Applied to Complex ODE System with Hidden Hierarchy Application to Turbocharger Engine Model

Abstract In this paper, we present the concept of singularly perturbed vector field (SPVF) method and its application to spark ignition turbocharger engine. Given a mathematical/physical model, which consist of hidden multi-scale variables, the SPVF methods transfer (using the change of coordinates) and decompose such system to fast and slow subsystems. This decomposition enables one to apply different asymptotic methods such as the method of the integral manifold, homotopy analysis method, singular perturbation method, etc. The resulting subsystem enables us to understand better the complex system and to simplify the calculations. In addition, we investigated the stability of the equilibrium points of the model. This analysis has been done due to the SPVF method which reduces the complexity of the mathematical model.

Energy-preserving methods on Riemannian manifolds

The energy preserving discrete gradient methods are generalized to finite-dimensional Riemannian manifolds by definition of a discrete approximation to the Riemannian gradient, a retraction, and a coordinate center function. The resulting schemes are intrinsic and do not depend on a particular choice of coordinates, nor on embedding of the manifold in a Euclidean space. Generalizations of well-known discrete gradient methods, such as the average vector field method and the Itoh--Abe method are obtained. It is shown how methods of higher order can be constructed via a collocation-like approach. Local and global error bounds are derived in terms of the Riemannian distance function and the Levi-Civita connection. Some numerical results on spin system problems are presented.

Global Solution for Massive Maxwell‐Klein‐Gordon Equations

We derive the asymptotic properties of the mMKG system (Maxwell coupled with a massive Klein‐Gordon scalar field) in the exterior of the domain of influence of a compact set. This complements the previous well‐known results, restricted to compactly supported initial conditions, based on the so‐called hyperboloidal method. That method takes advantage of the commutation properties of the Maxwell and Klein‐Gordon equations with the generators of the Poincaré group to resolve the difficulties caused by the fact that they have, separately, different asymptotic properties. Though the hyperboloidal method is very robust and applies well to other related systems, it has the well‐known drawback of requiring compactly supported data. In this paper we remove this limitation based on a further extension of the vector field method adapted to the exterior region. Our method applies, in particular, to nontrivial charges. The full problem can then be treated by patching together the new estimates in the exterior with the hyperboloidal ones in the interior. This purely physical space approach introduced here maintains the robust properties of the old method and can thus be applied to other situations such as the coupled Einstein Klein‐Gordon equation. © 2019 Wiley Periodicals, Inc.

Long-time momentum and actions behaviour of energy-preserving methods for semi-linear wave equations via spatial spectral semi-discretisations

It is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semi-linear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. The results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First, a multi-frequency modulated Fourier expansion of the AAVF method is constructed, and then two almost-invariants of the modulation system are derived.

Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation

In $n \geq 1$ spatial dimensions, we study the Cauchy problem for a quasilinear transport equation coupled to a quasilinear symmetric hyperbolic subsystem of a rather general type. For an open set (relative to a suitable Sobolev topology) of regular initial data that are close to the data of a simple plane wave, we give a sharp, constructive proof of shock formation in which the transport variable remains bounded but its first-order Cartesian coordinate partial derivatives blow up in finite time. Moreover, we prove that the singularity does not propagate into the symmetric hyperbolic variables: they and their first-order Cartesian coordinate partial derivatives remain bounded, even though they interact with the transport variable all the way up to its singularity. The formation of the singularity is tied to the finite-time degeneration, relative to the Cartesian coordinates, of a system of geometric coordinates adapted to the characteristics of the transport operator. Two crucial features of the proof are that relative to the geometric coordinates, all solution variables remain smooth, and that the finite-time degeneration coincides with the intersection of the characteristics. Compared to prior shock formation results in more than one spatial dimension, in which the blowup occurred in solutions to wave equations, the main new features of the present work are: i) we develop a theory of nonlinear geometric optics for transport operators, which is compatible with the coupling and which allows us to implement a quasilinear geometric vectorfield method, even though the regularity properties of the corresponding eikonal function are less favorable compared to the wave equation case and ii) we allow for a full quasilinear coupling, i.e., the principal coefficients in all equations are allowed to depend on all solution variables.

A New Formulation of the 3D Compressible Euler Equations with Dynamic Entropy: Remarkable Null Structures and Regularity Properties

We derive a new formulation of the 3D compressible Euler equations exhibiting remarkable null structures and regularity properties. Our results hold for an arbitrary equation of state (which yields the pressure in terms of the density and the entropy) in non-vacuum regions where the speed of sound is positive. Our work here is an extension of our prior joint work with J. Luk, in which we derived a similar new formulation in the special case of a barotropic fluid, that is, when the equation of state depends only on the density. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity (defined to be vorticity divided by density), transport equations for the entropy and its gradient, and some additional transport–divergence–curl-type equations involving special combinations of the derivatives of the solution variables. The good geometric structures in the equations allow one to use the full power of the vectorfield method in treating the “wave part” of the system. In a forthcoming application, we will use the new formulation to give a sharp, constructive proof of finite-time shock formation, tied to the intersection of acoustic “wave characteristics,” for solutions with nontrivial vorticity and entropy at the singularity. In the present article, we derive the new formulation and provide an overview of the central role that it plays in the proof of shock formation. Although the equations are significantly more complicated than they are in the barotropic case, they enjoy many of same remarkable features, including: (i) all derivative-quadratic inhomogeneous terms are null forms relative to the acoustical metric, which is the Lorentzian metric driving the propagation of sound waves and (ii) the transport–divergence–curl-type equations allow one to show that the entropy is one degree more differentiable than the velocity and that the vorticity is exactly as differentiable as the velocity, assuming that the initial data enjoy the same gain in regularity. This represents a gain of one derivative compared to standard estimates. This gain of a derivative, which seems to be new for the entropy, is essential for closing the energy estimates in our forthcoming proof of shock formation, since the second derivatives of the entropy and the first derivatives of the vorticity appear as inhomogeneous terms in the wave equations.

Addressing Unmodeled Path-Following Dynamics via Adaptive Vector Field: A UAV Test Case

The actual performance of model-based path-following methods for unmanned aerial vehicles (UAVs) shows considerable dependence on the wind knowledge and on the fidelity of the dynamic model used for design. This study analyzes and demonstrates the performance of an adaptive vector field (VF) control law which can compensate for the lack of knowledge of the wind vector and for the presence of unmodeled course angle dynamics. Extensive simulation experiments, calibrated on a commercial fixed-wing UAV and proven to be realistic, show that the new VF method can better cope with uncertainties than its standard version. In fact, while the standard VF approach works perfectly for ideal first-order course angle dynamics (and perfect knowledge of the wind vector), its performance degrades in the presence of unknown wind or unmodeled course angle dynamics. On the other hand, the estimation mechanism of the proposed adaptive VF effectively compensates for wind uncertainty and unmodeled dynamics, sensibly reducing the path-following error as compared to the standard VF.

Efficient energy‐preserving scheme of the three‐coupled nonlinear Schrödinger equation

An energy‐preserving scheme is proposed for the three‐coupled nonlinear Schrödinger (T‐CNLS) equation. The T‐CNLS equation is rewritten into the classical Hamiltonian form. Then the spatial variable is discretized by using high‐order compact method to convert it into a finite‐dimensional Hamiltonian system. Next, a second‐order averaged vector field (AVF) method is employed in time which results in an energy‐preserving scheme. Some theoretical results such as convergence are investigated. In addition, it provides some numerical examples to illustrate the robustness and reliability of the theoretical results. It also explores the role of the parameters in the model and initial condition on the wave propagation.

STOCHASTIC PARTITIONED AVERAGED VECTOR FIELD METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH A CONSERVED QUANTITY

In this paper, stochastic differential equations in the Stratonovich sense with a conserved quantity are considered. A stochastic partitioned averaged vector field method is proposed and analyzed. We prove this numerical method is able to preserve the conserved quantity of the original system. Then the convergence analysis is carried out in detail and we derive the method is convergent with order 1 in the mean-square sense. Finally some numerical examples are reported to verify the effectiveness and flexibility of the proposed method.

Structure preserving reduced order modeling for gradient systems

Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg–Landau equation and Swift–Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state.

A Vector Field Approach to Almost-Sharp Decay for the Wave Equation on Spherically Symmetric, Stationary Spacetimes

We present a new vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary and asymptotically flat spacetimes. Specifically, we derive a new hierarchy of higher-order weighted energy estimates by employing appropriate commutator vector fields. In cases where an integrated local energy decay estimate holds, like in the case of sub-extremal Reissner–Nordstrom black holes, this hierarchy leads to almost-sharp global energy and pointwise time-decay estimates with decay rates that go beyond those obtained by the traditional vector field method. Our estimates play a fundamental role in our companion paper where precise late-time asymptotics are obtained for linear scalar fields on such backgrounds.

Conservative or dissipative quasi-interpolation method for evolutionary partial differential equations

Based on quasi-interpolation, we propose a new meshless conservative or dissipative method for nonlinear time-dependent partial differential equations. Using the method of lines, we first discretize the equation in space with the quasi-interpolation method, then employ the average vector field method in time discretization to derive the final numerical scheme. The method not only inherits the conservation or dissipation property of the equation but also has the meshless feature since we use the nonuniform grids in spatial discretization. Several numerical examples are presented to demonstrate the effectiveness of the proposed method.