Nonlinear Hyperbolic Partial Differential Equations Research Articles

A1D–0D–3Dcoupled model for simulating blood flow and transport processes in breast tissue

In this work, we present mixed dimensional models for simulating blood flow and transport processes in breast tissue and the vascular tree supplying it. These processes are considered, to start from the aortic inlet to the capillaries and tissue of the breast. Large variations in biophysical properties and flow conditions exist in this system necessitating the use of different flow models for different geometries and flow regimes. In total, we consider four different model types. First, a system of 1D nonlinear hyperbolic partial differential equations (PDEs) is considered to simulate blood flow in larger arteries with highly elastic vessel walls. Second, we assign 1D linearized hyperbolic PDEs to model the smaller arteries with stiffer vessel walls. The third model type consists of ODE systems (0D models). It is used to model the arterioles and peripheral circulation. Finally, homogenized 3D porous media models are considered to simulate flow and transport in capillaries and tissue within the breast volume. Sink terms are used to account for the influence of the venous and lymphatic systems. Combining the four model types, we obtain two different 1D-0D-3D coupled models for simulating blood flow and transport processes: The first model results in a fully coupled 1D-0D-3D model covering the complete path from the aorta to the breast combining a generic arterial network with a patient specific breast network and geometry. The second model is a reduced one based on the separation of the generic and patient specific parts. The information from a calibrated fully coupled model is used as inflow condition for the patient specific sub-model allowing a significant computational cost reduction. Several numerical experiments are conducted to calibrate the generic model parameters and to demonstrate realistic flow simulations compared to existing data on blood flow in the human breast and vascular system. Moreover, we use two different breast vasculature and tissue data sets to illustrate the robustness of our reduced sub-model approach.

Error estimates for DeepONets: a deep learning framework in infinite dimensions

Abstract DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite-dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In particular, we extend the universal approximation property of DeepONets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ordinary differential equation, an elliptic partial differential equation (PDE) with variable coefficients and nonlinear parabolic and hyperbolic PDEs. While the approximation of arbitrary Lipschitz operators by DeepONets to accuracy $\epsilon $ is argued to suffer from a ‘curse of dimensionality’ (requiring a neural networks of exponential size in $1/\epsilon $), in contrast, for all the above concrete examples of interest, we rigorously prove that DeepONets can break this curse of dimensionality (achieving accuracy $\epsilon $ with neural networks of size that can grow algebraically in $1/\epsilon $).Thus, we demonstrate the efficient approximation of a potentially large class of operators with this machine learning framework.

Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator

SUMMARY Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used due to their efficiency in numerically solving the eikonal equation. However, these methods suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity, irregular surface topography and complex phase velocity fields. This study presents a new method for the solution of the eikonal equation by employing the peridynamic differential operator (PDDO). The PDDO provides the non-local form of the eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional non-locality. The operator is immune to discontinuities in the form of sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the strength of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The robustness of this approach is established by considering different types of eikonal equations on complex velocity models in anisotropic media. The examples demonstrate its unconditional numerical stability and results compare well with the reference solutions.

On the solution of hyperbolic equations using the peridynamic differential operator

Numerical solution of hyperbolic differential equations, such as the advection equation, poses challenges. Classically, this issue has been addressed by using a scheme known as the upwind scheme. It simply invokes more points from the upwind side of the flow stream when calculating derivatives. This study presents a generalized upwind scheme, referred to as directional nonlocality, for the numerical solution of linear and nonlinear hyperbolic Partial Differential Equations (PDEs) using the peridynamic differential operator (PDDO). The PDDO provides the nonlocal form of the differential equations by introducing an internal length parameter (horizon) and a weight function. The weight function controls the degree of interaction among the points within the horizon. A modification to the weight function, i.e., upwinded-weight function, accounts for directional nonlocality along which information travels. This modification results in a stable PDDO discretization of hyperbolic PDEs. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering time dependent linear and nonlinear hyperbolic equations as well as the time invariant Eikonal equation. Numerical stability is ensured for the linear advection equation and the PD solutions compare well with the analytical/reference solutions.

Second-Order Flows for Computing the Ground States of Rotating Bose-Einstein Condensates

Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modelled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. This is judged by the fact that larger size of time steps can be applied in explicit methods, and much smaller number of time steps are needed in both explicit and semi-implicit discretizations, while every time step encounters almost the same computational complexity in comparison with the respective gradient flow type approaches. Rich and detailed numerical examples are documented for verification and comparison.

Conservation Laws -- a Source for Distortionless Propagation and Time Delays

Since the very first paper of J. Bernoulli in 1728, a connection exists between initial boundary value problems for hyperbolic Partial Differential Equations (PDE) in the plane (with a single space coordinate accounting for wave propagation) and some associated Functional Equations (FE). From the point of view of dynamics and control (to be specific, of dynamics for control) both type of equations generate dynamical and controlled dynamical systems. The functional equations may be difference equations (in continuous time), delay-differential (mostly of neutral type) or even integral/integro-differential. It is possible to discuss dynamics and control either for PDE or FE since both may be viewed as self contained mathematical objects. A more recent topic is control of systems displaying conservation laws. Conservation laws are described by nonlinear hyperbolic PDE belonging to the class ``lossless'' (conservative); their dynamics and control theory is well served by the associated energy integral. It is however not without interest to discuss association of some FE. Lossless implies usually distortionless propagation hence one would expect here also lumped time delays. The paper contains some illustrating applications from various fields: nuclear reactors with circulating fuel, canal flows control, overhead crane, drilling devices, without forgetting the standard classical example of the nonhomogeneous transmission lines for distortionless and lossless propagation. Specific features of the control models are discussed in connection with the control approach wherever it applies.

A New Class of Coupled Systems of Nonlinear Hyperbolic Partial Fractional Differential Equations in Generalized Banach Spaces Involving the ψ–Caputo Fractional Derivative

The current study is devoted to investigating the existence and uniqueness of solutions for a new class of symmetrically coupled system of nonlinear hyperbolic partial-fractional differential equations in generalized Banach spaces in the sense of ψ–Caputo partial fractional derivative. Our approach is based on the Krasnoselskii-type fixed point theorem in generalized Banach spaces and Perov’s fixed point theorem together with the Bielecki norm, while Urs’s approach was used to prove the Ulam–Hyers stability of solutions of our system. Finally, some examples are provided in order to illustrate our theoretical results.

Event-triggered boundary control of a continuum model of highly re-entrant manufacturing systems

We design an event-triggered boundary controller for a continuum model of highly re-entrant manufacturing systems for which the influx rate of products is the controlled quantity. Mathematically, the governing equation of the process is a nonlinear hyperbolic partial differential equation with a nonlinearity arising from the nonlocal propagation speed and the positivity state constraint. The nonlinearity makes highly non-trivial the proof of well-posedness of the closed-loop system under ETC, and the design of the triggering mechanism requires novelty because standard triggering schemes proposed in the literature of linear PDEs do not work. The designed controller can potentially operate in networked control systems subject to limited information sharing resources. A Lyapunov argument is utilized to derive the boundary controller together with a feasible event generator that avoids the occurrence of Zeno behavior for the closed-loop system. The global stability estimate is established using the logarithmic norm of the state due to the system’s nonlinearity and positivity of the density. Furthermore, robustness of the proposed controller with respect to the sampling schedule and sampled-data stabilization results are established. Consistent simulation results that support the proposed theoretical statements are provided.

Error Estimates for Galerkin Method to a Type of Partial Differential Equations

Discuss Galerkin approximation to a type of second order nonlinear hyperbolic partial differential equations. Provide continuous time Galerkin approximation and fully discrete Galerkin approximations and derive optimal L2-error estimates for continuous time and fully discrete Calerkin approximations, separately.

A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws

The weighted essentially non-oscillatory (WENO) methods are popular and effective spatial discretization methods for nonlinear hyperbolic partial differential equations. Although these methods are formally first-order accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. In this paper, we propose a novel third-order numerical method for solving optimal control problems subject to scalar nonlinear hyperbolic conservation laws. It is based on the first-disretize-then-optimize approach and combines a discrete adjoint WENO scheme of third order with the classical strong stability preserving three-stage third-order Runge–Kutta method SSPRK3. We analyze its approximation properties and apply it to optimal control problems of tracking-type with non-smooth target states. Comparisons to common first-order methods such as the Lax–Friedrichs and Engquist–Osher method show its great potential to achieve a higher accuracy along with good resolution around discontinuities.

Solving free boundary problem for an initial cell layer in multispecies biofilm formation by Newton-Raphson method

The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.

A dynamic framework for a three phase hybrid flotation column

A comprehensive dynamic fundamental model with Danckwerts’ boundary conditions at the pulp/froth interface has been developed for a continuous hybrid flotation column that includes a mechanically agitated section at the bottom. The model accounts for three phases, namely the water, gas, and solid particles (hydrophobic and hydrophilic), in one space dimension. The modeling framework is based on three subsystems, including a well-mixed reactor and two plug-flow reactors representing pulp (collection) and froth (cleaning) zones that are interconnected through the boundaries. The model considers the micro-scale processes taking place in the column, such as bubble-particle attachment and bubble coalescence. The resulting mathematical model is a coupled set of nonlinear heterodirectional hyperbolic partial differential equations for the pulp and the froth and a set of ordinary differential equations for the well-mixed zone. The movement between phases are given by liquid upflow, liquid downflow, and gas upflow. An important parameter in this simulation is the bubble size, which directly affects the gas holdup, and consequently the distributions of all states, including the concentration of attached minerals through the column. Qualitative validation of the model is provided by testing its predictions against experimental data from the literature for two-phase systems. Finally, the effect of some important parameters such as gas flow rate, agitation, particle size on gas holdup, distributions of value minerals, grade, recovery, and attachment/detachment rates have been studied.

Draping woven sheets

Motivated by the manufacture of carbon fibre components, this paper considers the smooth draping of loosely woven fabric over rigid obstacles, both smooth and nonsmooth. The draped fabric is modelled as the continuum limit of a Chebyshev net of two families of short rigid rods that are freely pivoted at their joints. This approach results in a system of nonlinear hyperbolic partial differential equations whose characteristics are the fibres in the fabric. The analysis of this system gives useful information about the drapability of obstacles of many shapes and also poses interesting theoretical questions concerning well-posedness, smoothness and computability of the solutions. doi:10.1017/S144618112000019X

Output feedback control of two-lane traffic congestion

This paper develops output feedback boundary control to mitigate traffic congestion of an unidirectional two-lane freeway segment. The macroscopic traffic dynamics are described by the Aw–Rascle–Zhang (ARZ) model respectively for both the fast and slow lanes. The traffic density and velocity of each of the two lanes are governed by coupled 2 × 2 nonlinear hyperbolic partial differential equations (PDEs). Lane-changing interactions between the two lanes lead to exchanging source terms between the two pairs second-order PDEs. Therefore, we are dealing with 4 × 4 nonlinear coupled hyperbolic PDEs. Based on driver’s preference for the slow and fast lanes, a reference system of lane-specific uniform steady states in congested traffic is chosen. To stabilize traffic densities and velocities of both lanes to the steady states, two distinct variable speed limits (VSLs) are applied at outlet boundary, controlling the traffic velocity of each lane. Using backstepping transformation, we map the coupled heterodirectional hyperbolic PDE system into a cascade target system, in which the traffic oscillations are damped out through actuation of the velocities at the downstream boundary. Two full-state feedback boundary control laws are developed. We also design a collocated boundary observer for state estimation with sensing of the densities at the outlet. Output feedback boundary controllers are obtained by combining the collocated observer and full-state feedback controllers. The finite time convergence to equilibrium is achieved and a performance improvement in fuel consumption, drivers’ comfort and total travel time is demonstrated with the proposed output feedback controllers for some parameter choice, compared with the open-loop system. The proposed output feedback design is also validated for different congested traffic scenarios and compared with one-lane backstepping and PI control designs.

Inverse initial boundary value problem for a non-linear hyperbolic partial differential equation

In this article we are concerned with an inverse initial boundary value problem for a non-linear wave equation in space dimension n ⩾ 2. In particular we consider the so called interior determination problem. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear wave equation with a time independent potential. For any small solution u = u(t, x) of our non-linear wave equation which is the perturbation of linear wave equation with time-independent potential perturbed by a divergence with respect to (t, x) of a vector whose components are quadratics with respect to ∇ t,x u(t, x). By ignoring the terms with smallness O(|∇ t,x u(t, x)|3), we will show that we can uniquely determine the potential and the coefficients of these quadratics by many boundary measurements at the boundary of the spacial domain over finite time interval and the final overdetermination at t = T. In other words, our measurement is given by the so-called the input-output map (see ()).

Spatio-stochastic adaptive discontinuous Galerkin methods

We introduce a goal-oriented, adaptive framework for the uncertainty quantification of systems modeled by stochastically parametrized nonlinear hyperbolic and convection-dominated partial differential equations. Of particular interest are conservation laws in aerodynamics that may have a small number of stochastic parameters but exhibit strong nonlinearity and a wide range of scales. Our framework exploits localized structure in the spatio-parameter space to enable rapid, reliable uncertainty quantification for output quantities of interest. Our formulation comprises the following technical components: (i) a discontinuous Galerkin finite element method, which provides stability for convection-dominated problems; (ii) element-wise polynomial chaos expansions, which capture the parametric dependence of the solution in a way amenable to adaptation; (iii) the dual-weighted residual method, which provides global and element-wise error estimates for quantities of interest; and (iv) a projection-based anisotropic error indicator along with the associated adaptation mechanics that can detect and refine strongly directional features in the physical and/or parameter spaces simultaneously in an efficient manner. Both the spatial and stochastic discretization errors are controlled through the adaptive refinement of the spatial mesh or polynomial chaos expansion degree based on these anisotropic error indicators. We analyze stability, approximation properties, and a priori error bounds of the spatio-stochastic adaptive method. We finally demonstrate the effectiveness of our formulation for engineering-relevant transonic turbulent aerodynamics problems with uncertainties in flow conditions and turbulence parameters.

DRAPING WOVEN SHEETS

AbstractMotivated by the manufacture of carbon fibre components, this paper considers the smooth draping of loosely woven fabric over rigid obstacles, both smooth and nonsmooth. The draped fabric is modelled as the continuum limit of a Chebyshev net of two families of short rigid rods that are freely pivoted at their joints. This approach results in a system of nonlinear hyperbolic partial differential equations whose characteristics are the fibres in the fabric. The analysis of this system gives useful information about the drapability of obstacles of many shapes and also poses interesting theoretical questions concerning well-posedness, smoothness and computability of the solutions.

Nonlinear Hyperbolic Problems: Modeling, Analysis, and Numerics

The workshop gathered together leading international experts, as well as most promising young researchers, working on the modelling, the mathematical analysis, and the numerical methods for nonlinear hyperbolic partial differential equations (PDEs). The meeting focussed on addressing outstanding issues and identifying promising new directions in all three fields, i.e. modelling, analysis, and numerical discretization. Key questions settled around the lack of well-posedness theories for multidimensional systems of conservation laws and the use of hyperbolic modelling beyond the classical topic of gas dynamics. A focal point in numerics has been the discretization of random evolutions and uncertainty quantification. Equally important, new multi-scale methods and schemes for asymptotic regimes have been considered.

Free boundary approach to modelling multispecies biofilms

The work presents the free boundary approach to modelling the formation and growth of multispecies biofilms with special attention to recent new problems. The mathematical free boundary value problem is formalized by a system of nonlinear hyperbolic partial differential equations that governs the multispecies biofilm growth and a system of semi-linear parabolic partial differential equations that governs the substrate diffusion within the biofilm. The biofilm thickness is the free boundary of the problem. Its evolution over time is governed by a differential equation that involves the growth velocity of microbial mass, attachment and detachment. A generalized model is introduced to include the possibility that the planktonic bacterial cells could penetrate an already constituted biofilm and colonize the regions where favorable conditions for their growth are found. The new process is governed by parabolic equations mutually connected with all the others. Moreover, the biological process of the initial biofilm formation is modelled. The proposed model does not require any initial conditions as the biofilm thickness is zero and the species composition is modelled according to the environmental conditions. The related mathematical problem is interesting since the free boundary is a space-like line and its initial value is zero. This last problem is also discussed numerically. Since the initial biofilm thickness is zero, the free boundary domain cannot be transformed to a rectangular domain and most numerical methods cannot be applied. It is shown that the method of characteristics, successively used for the qualitative analysis, still works for the numerical solutions.

PDE Traffic Observer Validated on Freeway Data

This article develops a boundary observer for the estimation of congested freeway traffic states based on the Aw-Rascle-Zhang (ARZ) partial differential equations (PDEs) model. Traffic state estimation refers to the acquisition of traffic state information from partially observed traffic data. This problem is relevant for freeway due to its limited accessibility to real-time traffic information. We propose a model-driven approach in which the estimation of aggregated traffic states in a freeway segment is obtained simply from the boundary measurement of flow and velocity without knowledge of the initial states. The macroscopic traffic dynamics is represented by the ARZ model, a 2 ×2 coupled nonlinear hyperbolic PDEs for traffic density and velocity. Using the PDE backstepping method, we construct a boundary observer consisting of a copy of the nonlinear plant with output injections from boundary measurement errors. The exponential stability of the estimation error system in the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> norm and finite-time convergence to zero is guaranteed. Numerical simulation and data validation are conducted to validate the boundary observer design with vehicle trajectory data.