Hyperbolic-parabolic Partial Differential Equations Research Articles

A fully implicit edge/face centered discontinuous Galerkin / mixed finite element scheme for the advection-dispersion equation

Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection – diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: (i) it lacks efficiency, as two systems with different unknowns are solved at each time step, (ii) it induces errors generated by the splitting, (iii) it can be CPU wise-expensive because of the CFL constraint, and (iv) it cannot be employed for steady-state transport simulations.To overcome these difficulties, we develop in this work a fully implicit edge/face centered DG-MFE formulation where the two methods share the same unknowns. In this formulation, the DG method is developed on lumping regions associated with the mesh edges/faces instead of mesh elements. Thus, the traces of concentration at mesh edges/faces, which are the Degrees Of Freedom (DOF) of the hybrid-MFE, are also part of the DOFs of the DG. The temporal discretization is based on the Crank–Nicolson method for both advection and dispersion. Numerical tests are performed to validate the new scheme by comparison against an analytical solution and to show its ability to handle steady-state transport simulations.The procedure is developed for 2D triangular meshes but can easily be extended to other 2D and 3D shape elements.

Well posedness analysis of a parabolic-hyperbolic free boundary problem for necrotic tumors growth

In this paper, we study a free boundary problem of tumor growth with necrotic core. The model is a parabolic-hyperbolic partial differential equations, which is composed of three first-order nonlinear hyperbolic equations, a parabolic equation and an ordinary differential equation. First, we obtained the approximation model by polishing the Heaviside function, and then proved the existence and uniqueness of the solution of the approximation model. In addition, we improved the regularity of solution of the approximate problem by using the characteristic curves method, and finally proved the global existence of the weak solution of the original problem by the convergence.

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

We treat infinite horizon optimal control problems by solving the associated stationary Hamilton-Jacobi-Bellman (HJB) equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the HJB is non linear and suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear, hyperbolic PDEs. These equations remain the computational bottleneck due to their high dimensions. By the method of characteristics these linearized HJB equations can be reformulated via the Koopman operator in the spirit of dynamic programming. The resulting operator equations are solved using a minimal residual method. To overcome numerical infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors), and multi-polynomials, together with high dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidences are given.

The difference between semi-continuum model and Richards’ equation for unsaturated porous media flow

Semi-continuum modelling of unsaturated porous media flow is based on representing the porous medium as a grid of non-infinitesimal blocks that retain the character of a porous medium. This approach is similar to the hybrid/multiscale modelling. Semi-continuum model is able to physically correctly describe diffusion-like flow, finger-like flow, and the transition between them. This article presents the limit of the semi-continuum model as the block size goes to zero. In the limiting process, the retention curve of each block scales with the block size and in the limit becomes a hysteresis operator of the Prandtl-type used in elasto-plasticity models. Mathematical analysis showed that the limit of the semi-continuum model is a hyperbolic-parabolic partial differential equation with a hysteresis operator of Prandl’s type. This limit differs from the standard Richards’ equation, which is a parabolic equation and is not able to describe finger-like flow.

A new analysis of Galerkin Legendre spectral methods for coupled hyperbolic/parabolic system arising in unsteady MHD flow of Maxwell fluid and numerical simulation

In this paper, a high-order accurate numerical method for two-dimensional coupled hyperbolic and parabolic partial differential system arising in unsteady magnetohydrodynamics (MHD) flow of upper-convected Maxwell fluid is presented. We apply Galerkin-Legendre spectral method for discretizing spatial derivatives for MHD-M (MHD flow of Maxwell fluid) equations and then Crank-Nicolson scheme is used for time derivatives. Optimal a priori error bound is derived in the L2 and H1 norm for the semidiscrete formulation. We then proceed to discuss an instability mechanism in polynomial spectral methods and show that filtering suffices to ensure stability. Optimal error bound is derived between the semi-analytical solutions and filtered semi-analytical solutions. The results are illustrated by computational experiments. Furthermore, these results indicate that the proposed method is simple, accurate, efficient, and stable for solving various MHD-M and MHD flow problems or in general coupled hyperbolic-parabolic partial differential equations.

A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.

High order difference schemes using the local anisotropic basis function method

Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers from a lack of consistency. High order, consistent, and local (using compact computational stencils) mesh-free methods are particularly desirable.Here we present a novel framework for generating local high order difference operators for arbitrary node distributions, referred to as the Local Anisotropic Basis Function Method (LABFM). Weights are constructed from linear sums of anisotropic basis functions (ABFs), chosen to ensure exact reproduction of polynomial fields up to a given order. The ABFs are based on a fundamental Radial Basis Function (RBF), and the choice of fundamental RBF has small effect on accuracy, but influences stability. LABFM is able to generate high order difference operators with compact computational stencils (4th order with N≈25 nodes, 8th order with N≈60 nodes in two dimensions). At domain boundaries (with incomplete support) LABFM automatically provides one-sided differences of the same order as the internal scheme, up to 4th order. We use the method to solve elliptic, parabolic and mixed hyperbolic-parabolic partial differential equations (PDEs), showing up to 8th order convergence. The inclusion of hyperviscosity is straightforward, and can effectively provide stability when solving hyperbolic problems.LABFM is a promising new mesh-free method for the numerical solution of PDEs in complex geometries. The method is highly scalable, and for Eulerian schemes, the computational efficiency is competitive with RBF-FD for a given accuracy. A particularly attractive feature is that in the low order limit, LABFM collapses to Smoothed Particle Hydrodynamics (SPH), and there is potential for Arbitrary Lagrangian-Eulerian schemes with natural adaptivity of resolution and accuracy.

Model-Free Backward and Forward Nonlinear PDEs for Implied Volatility

The authors derive backward and forward nonlinear partial differential equations that govern the implied volatility of a contingent claim whenever the latter is well defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. The authors also discuss suitable initial and boundary conditions for those partial differential equations. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach. TOPICS:Volatility measures, options Key Findings • In this article, we derive backward and forward quasilinear parabolic partial differential equations (PDEs) that govern the implied volatility of a contingent claim whenever the latter is well defined. Alternatively, we have derived a forward nonlinear hyperbolic PDE of the first order, which also governs evolution of the implied volatility in (K, T, Z) space. We discuss suitable initial and boundary conditions for those PDEs. • We develop an iterative numerical method to solve the PDEs by using a finite-difference approach. The method is of second order of approximation in both space and time, is unconditionally stable, and preserves positivity of the solution. • Using this method, we compute the PDE implied volatility and find that our intuition behind the main idea of the article is correct. In other words, performance of the finite-difference solver exceeds that of the traditional approach by factor of 40. However, this result is subject to some details, which are highlighted in the article.

Response of a spherical cavity in a fully-coupled thermo-poro-elastodynamic medium by cell-adaptive second-order central high resolution schemes

In this study, fully coupled thermo-poroelastic saturated media are simulated by a grid/cell adaptive central high resolution scheme. The central method corresponds to the second order Kurganov–Tadmor (KT) scheme working on adapted cells with the total variation diminishing (TVD) stability condition. The coupled equations include motion, fluid flow, heat flow, continuity condition, and a constitutive equation. The grid/cell adaptation is performed by the interpolating wavelet transform in the multiresolution framework to capture fine scale responses and to obtain a computationally effective solver. With respect to the use of central schemes, the coupled equations should be re-expressed as a system of coupled first-order hyperbolic-parabolic partial differential equations (PDEs) with possible source (load) terms. The system is initially derived in the Cartesian coordinate system, and it is subsequently modified to consider a spherical cavity in isotropic, symmetric, and saturated media in the spherical coordinate system. It is assumed that the cavity boundary is subjected to sudden time-dependent thermal/mechanical sources. Discontinuous propagating fronts develop in the media due to the aforementioned loading. It is challenging to handle these solutions with numerical methods, and special attention is required to prevent/control numerical dispersion and dissipation. Hence, as previously mentioned, adaptive central high resolution schemes are employed in the present study.

An efficient variational iteration transform method for solving nonlinear parabolic-hyperbolic partial differential equations

An efficient variational iteration transform method for solving nonlinear parabolic-hyperbolic partial differential equations

Output and error feedback regulator designs for linear infinite-dimensional systems

This manuscript addresses the output regulation problem for linear distributed parameter systems (DPSs) with bounded input and unbounded output operators. We introduce novel methods for the design of the output feedback and error feedback regulators. In the output feedback regulator design, the measurements available for the regulator do not belong to the set of controlled outputs. The proposed output feedback regulator with the injection of the measurement ym(t) and reference yr(t) can realize both the plant and the exosystem states estimation, disturbance rejection and reference signal tracking, simultaneously. Moreover, new design approach provides an alternative choice for seeking the output injection gain in a traditional error feedback regulator design. The regulator parameters are easily configured to solve the output regulation problems, and to ensure the stability of the closed-loop systems. The results are demonstrated via computer simulation in two types of representative systems: the parabolic partial differential equation (PDE) system and the first order hyperbolic PDE system.

Transient Thermal Analysis of 3-D Integrated Circuits Packages by the DGTD Method

Since accurate thermal analysis plays a critical role in the thermal design and management of the 3-D system-level integration, in this paper, a discontinuous Galerkin time-domain (DGTD) algorithm is proposed to achieve this purpose. Such as the parabolic partial differential equation (PDE), the transient thermal equation cannot be directly solved by the DGTD method. To address this issue, the heat flux, as an auxiliary variable, is introduced to reduce the Laplace operator to a divergence operator. The resulting PDE is hyperbolic, which can be further written into a conservative form. By properly choosing the definition of the numerical flux used for the information exchange between neighboring elements, the hyperbolic thermal PDE can be solved by the DGTD together with the auxiliary differential equation. The proposed algorithm is a kind of element-level domain decomposition method, which is suitable to deal with multiscale geometries in 3-D integrated systems. To verify the accuracy and robustness of the developed DGTD algorithm, several representative examples are benchmarked.

Iterative learning control for a class of mixed hyperbolic-parabolic distributed parameter systems

This paper deals with the problem of iterative learning control algorithm for a class of mixed distributed parameter systems. Here, the considered distributed parameter systems are composed of mixed hyperbolic-parabolic partial differential equations. The domain of the system is divided into two parts in which the system is hyperbolic and parabolic, respectively, with transmission conditions at the interface. According to the characteristics of the systems, iterative learning control laws are proposed for such mixed hyperbolic-parabolic distributed parameter systems based on P-type learning scheme. Using the contraction mapping method, it is shown that the scheme can guarantee the output tracking errors on L 2 space converge along the iteration axis. A simulation example illustrates the effectiveness of the proposed method.

A class of hyperbolic-parabolic coupled systems applied to image restoration

This paper considers a coupled system with a new kind of hyperbolic-parabolic partial differential equations based on image restoration. We show that this system has global dissipative solutions under Dirichlet boundary conditions and initial condition. Meanwhile, an experimental approach is given to show the efficiency of this kind of model.

Traveling waves in Hall-magnetohydrodynamics and the ion-acoustic shock structure

Hall-MHD is a mixed hyperbolic-parabolic partial differential equation that describes the dynamics of an ideal two fluid plasma with massless electrons. We study the only shock wave family that exists in this system (the other discontinuities being contact discontinuities and not shocks). We study planar travelling wave solutions and we find solutions with discontinuities in the hydrodynamic variables, which arise due to the presence of real characteristics in Hall-MHD. We introduce a small viscosity into the equations and use the method of matched asymptotic expansions to show that solutions with a discontinuity satisfying the Rankine-Hugoniot conditions and also an entropy condition have continuous shock structures. The lowest order inner equations reduce to the compressible Navier-Stokes equations, plus an equation which implies the constancy of the magnetic field inside the shock structure. We are able to show that the current is discontinuous across the shock, even as the magnetic field is continuous, and that the lowest order outer equations, which are the equations for travelling waves in inviscid Hall-MHD, are exactly integrable. We show that the inner and outer solutions match, which allows us to construct a family of uniformly valid continuous composite solutions that become discontinuous when the diffusivity vanishes.

Slow manifolds for multi-time-scale stochastic evolutionary systems

This article deals with invariant manifolds for infinite dimensional random dynamical systems with different time scales. Such a random system is generated by a coupled system of fast-slow stochastic evolutionary equations. Under suitable conditions, it is proved that an exponentially tracking random invariant manifold exists, eliminating the fast motion for this coupled system. It is further shown that if the scaling parameter tends to zero, the invariant manifold tends to a slow manifold which captures long time dynamics. As examples the results are applied to a few systems of coupled parabolic-hyperbolic partial differential equations, coupled parabolic partial differential-ordinary differential equations, and coupled hyperbolic-hyperbolic partial differential equations.

Boundary estimation for a diffusion-reaction PDE driven by an unknown periodic Neumann input

In this paper, we design exponentially convergent observers for a class of parabolic partial differential equations (PDEs) driven by an unknown periodic Neumann input with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable system. Observer gains (output injection function) are shown to satisfy a well-posed hyperbolic PDE. Moreover, the observer gains are obtained in closed form. Simulation results are presented.

A Closed-Form Full-State Feedback Controller for Stabilization of 3D Magnetohydrodynamic Channel Flow

We present a boundary feedback law that stabilizes the velocity, pressure, and electromagnetic fields in a magnetohydrodynamic (MHD) channel flow. The MHD channel flow, also known as Hartmann flow, is a benchmark for applications such as cooling, hypersonic flight, and propulsion. It involves an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier–Stokes equations and a Poisson equation for the electric potential under the MHD approximation in a low magnetic Reynolds number regime. This model is unstable for large Reynolds numbers and is stabilized by actuation of velocity and the electric potential at only one of the walls. The backstepping method for stabilization of parabolic partial differential equations (PDEs) is applied to the velocity field system written in appropriate coordinates. Control gains are computed by solving a set of linear hyperbolic PDEs. Stabilization of nondiscretized 3D MHD channel flow has so far been an open problem.

Modeling and numerical simulation of secondary settlers: A Method of Lines strategy

In this paper, attention is focused on a parabolic partial differential equation (PDE) modeling sedimentation in a secondary settler and the proper formulation of the problem boundary conditions (i.e., the conditions prevailing at the feed, clear water and sludge outlets). The presence of a diffusion term in the equation not only allows the reproduction of experimental observations, as reported in a number of works, but also makes the numerical solution of the initial-boundary value problem significantly easier than the original conservation law (which is a nonlinear hyperbolic PDE problem requiring advanced numerical techniques). A Method of Lines (MOL) solution strategy is then proposed, based on the use of finite differences or spectral methods, and on readily available time integrators. The efficiency and flexibility of the general procedure are demonstrated with various numerical simulation results.

Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder

Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder