Higher-order Spatial Derivatives Research Articles

Mathematical modeling and qualitative analysis of viscoelastic conductive fluids

We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier–Stokes–Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

An ultraweak-local discontinuous Galerkin method for PDEs with high order spatial derivatives

In this paper, we develop a new discontinuous Galerkin method for solving several types of partial differential equations (PDEs) with high order spatial derivatives. We combine the advantages of a local discontinuous Galerkin (LDG) method and the ultraweak discontinuous Galerkin (UWDG) method. First, we rewrite the PDEs with high order spatial derivatives into a lower order system, then apply the UWDG method to the system. We first consider the fourth order and fifth order nonlinear PDEs in one space dimension, and then extend our method to general high order problems and two space dimensions. The main advantage of our method over the LDG method is that we have introduced fewer auxiliary variables, thereby reducing memory and computational costs. The main advantage of our method over the UWDG method is that no internal penalty terms are necessary in order to ensure stability for both even and odd order PDEs. We prove the stability of our method in the general nonlinear case and provide optimal error estimates for linear PDEs for the solution itself as well as for the auxiliary variables approximating its derivatives. A key ingredient in the proof of the error estimates is the construction of the relationship between the derivative and the element interface jump of the numerical solution and the auxiliary variable solution of the solution derivative. With this relationship, we can then use the discrete Sobolev and Poincaré inequalities to obtain the optimal error estimates. The theoretical findings are confirmed by numerical experiments.

An effective pure meshfree method for 1D/2D time fractional convection-diffusion problems on irregular geometry

An easy implemented and effective pure meshfree method is first developed to solve the 1D/2D constant/variable-order time fractional convection-diffusion equation (TF-CDE) on non-regular domain with two boundary conditions in this paper. The proposed method (CSPH-FDM) is derived from that the finite difference scheme (FDM) for Caputo time fractional derivative and a corrected smoothed particle hydrodynamics (CSPH) without kernel derivative for spatial derivatives. In the proposed CSPH-FDM, the high-order spatial derivative is divided into multi first-order derivatives and solved continuously by the CSPH, the Neumann boundary condition can be accurately treated, the two distribution cases of the local refinement and irregular particles or the arbitrary irregular shape domain can be easily and effectively implemented by the CSPH. To demonstrate the validity and numerical convergent order of the proposed method, several 1D/2D analytical examples with local refinement and irregular particles distributions or on complex geometries are first investigated, in which a four-order derivate problem with Neumann boundary is also considered. Subsequently, the CSPH-FDM is extended to predict the solute moving process versus time by two TF-CDEs on different irregular domains and compared with other numerical results. All the numerical results show the flexible application ability and reliability of the present method.

Decay Rates of the Compressible Hall-MHD Equations for Quantum Plasmas

In this paper, decay rates of the compressible Hall-MHD equations for quantum plasmas in three-dimensional whole space are studied. By using a general energy method, the time decay rates for higher-order spatial derivatives of density, velocity and magnetic field are established when the initial perturbation belongs to $\dot{H}^{-s}$ with $0 \leq s < \frac{3}{2}$ .

Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term

An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.

Decay rates of the compressible Hall-magnetohydrodynamic model for quantum plasmas

In this paper, the compressible Hall-magnetohydrodynamic model for quantum plasmas in three-dimensional whole space is considered. By using the Lp-Lq estimates and the Fourier splitting method, if the initial perturbation is small in H4 × H3 × H3 and bounded in the L1 norm, we obtain the time decay rates for low- and higher-order spatial derivatives of density, velocity, and magnetic field.

On the well-posedness and temporal decay for the 3D generalized incompressible Hall-MHD system

In this paper, we prove a new result on the properties of decay character $$r^*$$ (see Lemma 2.6) and then show a small data global well-posedness result for three-dimensional generalized incompressible Hall-MHD system. In the end, through Fourier splitting method, the properties of decay character $$r^*$$ and mathematical induction, we study the decay rate of higher-order spatial and time derivatives of strong solutions to such system.

On the Cauchy problem of a sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures

We study the well-posedness and asymptotic behavior of solutions to the Cauchy problem of a three-dimensional sixth-order Cahn–Hilliard equation arising in oil-water-surfactant mixtures. First, by using the pure energy method and a standard continuity argument, we prove that there exists a unique global strong solution provided that the H 2 -norm of the initial data is sufficiently small. Moreover, we establish suitable negative Sobolev norm estimates and obtain the optimal decay rates of the higher-order spatial derivatives of the strong solution.

Optimal Large-Time Behavior of the Two-Phase Fluid Model in the Whole Space

We investigate the large-time behavior of strong solutions to a two-phase fluid model in the whole space $\mathbb R^3$. This model was first derived by Choi [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122] by taking the hydrodynamic limit from the Vlasov--Fokker--Planck/isentropic Navier--Stokes equations with strong local alignment forces. Under the assumption that the initial perturbation around an equilibrium state is sufficiently small, the global well-posedness issue has been established in [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122]. However, as indicated by Choi, the large-time behavior of these solutions has remained an open problem. In this article, we resolve this problem by proving convergence to its associated equilibrium with the optimal rate which is the same as that of the heat equation. Particularly, the optimal convergence rates of the higher-order spatial derivatives of the solutions are also obtained. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates. Our method is based on Hodge decomposition, low-frequency and high-frequency decomposition, delicate spectral analysis, and energy methods.

ADER Methods for Hyperbolic Equations with a Time-Reconstruction Solver for the Generalized Riemann Problem: the Scalar Case

The ADER approach to solve hyperbolic equations to very high order of accuracy has seen explosive developments in the last few years, including both methodological aspects as well as very ambitious applications. In spite of methodological progress, the issues of efficiency and ease of implementation of the solution of the associated generalized Riemann problem (GRP) remain the centre of attention in the ADER approach. In the original formulation of ADER schemes, the proposed solution procedure for the GRP was based on (i) Taylor series expansion of the solution in time right at the element interface, (ii) subsequent application of the Cauchy–Kowalewskaya procedure to convert time derivatives to functionals of space derivatives, and (iii) solution of classical Riemann problems for high-order spatial derivatives to complete the Taylor series expansion. For realistic problems the Cauchy–Kowalewskaya procedure requires the use of symbolic manipulators and being rather cumbersome its replacement or simplification is highly desirable. In this paper we propose a new class of solvers for the GRP that avoid the Cauchy–Kowalewskaya procedure and result in simpler ADER schemes. This is achieved by exploiting the history of the numerical solution that makes it possible to devise a time-reconstruction procedure at the element interface. Still relying on a time Taylor series expansion of the solution at the interface, the time derivatives are then easily calculated from the time-reconstruction polynomial. The resulting schemes are called ADER-TR. A thorough study of the linear stability properties of the linear version of the schemes is carried out using the von Neumann method, thus deducing linear stability regions. Also, via careful numerical experiments, we deduce stability regions for the corresponding non-linear schemes. Numerical examples using the present simplified schemes of fifth and seventh order of accuracy in space and time show that these compare favourably with conventional ADER methods. This paper is restricted to the one-dimensional scalar case with source term, but preliminary results for the one-dimensional Euler equations indicate that the time-reconstruction approach offers significant advantages not only in terms of ease of implementation but also in terms of efficiency for the high-order range schemes.

Data-driven discovery of partial differential equation models with latent variables.

In spatially extended systems, it is common to find latent variables that are hard, or even impossible, to measure with acceptable precision but are crucially important for the proper description of the dynamics. This substantially complicates construction of an accurate model for such systems using data-driven approaches. The present paper illustrates how physical constraints can be employed to overcome this limitation using the example of a weakly turbulent quasi-two-dimensional Kolmogorov flow driven by a steady Lorenz force with an unknown spatial profile. Specifically, the terms involving latent variables in the partial differential equations governing the dynamics can be eliminated at the expense of raising the order of that equation. We show that local polynomial interpolation combined with sparse regression can handle data on spatiotemporal grids that are representative of typical experimental measurement techniques such as particle image velocimetry. However, we also find that the reconstructed model is sensitive to measurement noise and trace this sensitivity to the presence of high-order spatial and/or temporal derivatives.

Adaptive damage localization based on locally perturbed dynamic equilibrium and hierarchical clustering

Pseudo-excitation (PE) method is a recently developed damage identification method for flexible structures containing components like beams, plates and shells. Characterized by the high-order spatial derivatives, the approach has been shown to feature a high sensitivity to local damage. However, two major issues, i.e. susceptibility to measurement noise and unknown material/structural properties, hamper its practical applications. To tackle these problems, an adaptive damage localization method is proposed for plate-type structures, which combines the PE method with hierarchical clustering. In the proposed method, a general dynamic equilibrium model, involving unknown material/structural properties, is statistically identified and further used for damage localization. Moreover, noise-induced effects are quantified by using a hierarchical clustering for performance assessment of damage localization and process optimization of spatial derivative estimation to achieve more accurate damage localization. Meanwhile, a data fusion scheme is developed to avoid blind inspection zones, thus enhancing the capability of damage localization. Both numerical and experimental studies of cantilever plates containing two damage zones are conducted to validate the feasibility and the effectiveness of the proposed adaptive damage localization method. Results demonstrate that the proposed method outperforms the traditional PE method in terms of detection accuracy and robustness.

Global Existence and Optimal Convergence Rates of Solutions for Three-Dimensional Electromagnetic Fluid System

In this article, we study the electromagnetic fluid system in three-dimensional whole space ℝ3. Under assumption of small initial data, we establish the unique global solution by energy method. Moreover, we obtain the time decay rates of the higher-order spatial derivatives of the solution by combining the Lp − Lq estimates for the linearized equations and an elaborate energy method when the L1-norm of the perturbation is bounded.

Decay rates of the compressible quantum magnetohydrodynamic model

In this paper, decay rates of the compressible viscous quantum magnetohydrodynamic model in three-dimensional whole space are studied. By using a general energy method, the time decay rates for higher-order spatial derivatives of density, velocity and magnetic field are established when the initial perturbation belongs to H˙−s with 0≤s<32.

Global existence and asymptotic stability of the fractional chemotaxis–fluid system in [formula omitted

In this paper, we consider a model arising from biology, consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in R3. We establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument. Moreover, the decay rates of the solutions and their higher-order spatial derivatives are established towards the equilibrium by introducing the negative Sobolev space Ḣ−s(0⩽s<32).

A rectangular staggered-grid finite-difference scheme with fourth-order temporal accuracy for pseudo-acoustic VTI modelling

ABSTRACTCurrent staggered-grid finite-difference (SFD) schemes for anisotropic wave equations can ensure only high-order spatial modelling accuracy, whereas the time discretisation accuracy is still of second-order. To improve temporal modelling accuracy, we develop a rectangular SFD scheme with fourth-order temporal accuracy to solve the first-order pseudo-acoustic wave equation in vertical transversely isotropic media. High-order temporal accuracy is achieved by applying the high-order temporal derivatives, which can be replaced with the high-order spatial derivatives to reduce the computational cost. The corresponding high-order finite-difference (FD) coefficients are generated by using Taylor-series expansion and least-squares. Dispersion and stability analyses indicate that our proposed method has higher accuracy and better stability than the conventional method. Several modelling examples confirm that our proposed scheme exhibits low temporal dispersion even for a large time step. Because of the high-order accuracy in time and space domains, our new SFD scheme allows for a larger time step and shorter operator length, which is more efficient.

The Boltzmann equation with frictional force for very soft potentials in the whole space

We develop a general energy method for proving the optimal time decay rates of the higher-order spatial derivatives of solutions to the Boltzmann-type and Landau-type systems in the whole space, for both hard potentials and soft potentials. With the help of this method, we establish the global existence and temporal convergence rates of solution near a given global Maxwellian to the Cauchy problem on the Boltzmann equation with frictional force for very soft potentials i.e. $ -3&lt;\gamma&lt;-2 $.

Long-time behavior of solutions for the compressible quantum magnetohydrodynamic model in $${\mathbb {R}}^3$$ R 3

In this paper, long-time behavior of solutions for the compressible viscous quantum magnetohydrodynamic model in three-dimensional whole space is studied. We establish the optimal time decay rates for higher-order spatial derivatives of density, velocity and magnetic field, which improve the work of Pu and Xu (Z Angew Math Phys 68:18, 2017).

High order integration factor methods for systems with inhomogeneous boundary conditions

Abstract Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential equations. Implicit integration method (IIF) method along with its compact form (cIIF), which treats spatial derivatives exactly and reaction terms implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reaction and spatial derivatives. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. The compact representation for IIF (cIIF) was introduced to save the computational cost and storage for this purpose. Another challenge is finding the matrix of high order space discretization, especially near the boundaries. In this paper, we extend IIF method to high order discretization for spatial derivatives through an example of reaction diffusion equation with fourth order accuracy, while the computational cost and storage are similar to the general second order cIIF method. The method can also be efficiently applied to deal with other types of partial differential equations with both homogeneous and inhomogeneous boundary conditions. Direct numerical simulations demonstrate the efficiency and accuracy of the approach.

Structural Intensity Analysis on Irregular Shells

A method is presented to assess the transmission path of vibration energy and to localize sources or sinks on shells with arbitrary shape, constant thickness, and isotropic material properties. The derived equations of the structural intensity (SI) are based on the Kirchhoff–Love postulates and are formulated in terms of displacements, Lamé parameters, principal curvatures, and their partial derivatives with respect to the principal curvilinear coordinates (PCC). To test the accuracy of the method, two numerical models of thin shells with nonuniform curvatures were developed. The coordinates, displacements, and principal curvature directions (PCDs) at the shell's outer surface were used to estimate the SI vector fields and the energy density at the shell's middle surface. The power estimated from the surface integral of the divergence of the SI over the source was compared to the actual power injected in the shell. The absolute error in both models did not exceed 1%, showing that, in theory, the method is able to handle the high-order spatial derivatives of the displacement and geometry data. The qualitative effect of varying the internal damping in the models on the energy transmission was also investigated.