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We develop an algorithm for the equivariant divergence formula of the unstable perturbation of unstable transfer operators, by progressively computing 2u many bounded vectors or covectors on one orbit, where u is the unstable dimension. Combining this with the nonintrusive adjoint shadowing algorithm, we obtain the fast adjoint response algorithm for the linear responses on physical measures of discrete-time uniformly hyperbolic systems. The cost is independent of the number of parameters. We demonstrate our algorithm on a 21-dimensional chaotic system, a system with stochastic noise, and systems with discontinuities. Unlike previous methods, our algorithms are not cursed by dimensionality or the sensitive dependence on initial conditions. [ABSTRACT FROM AUTHOR]

Strong stability (or monotonicity)-preserving time discretization schemes preserve the stability properties of the exact solution and have proved very useful in scientific and engineering computation, especially in solving hyperbolic partial differential equations. The main aim of this work is to further extend this to exponential stability-preserving numerical methods for a general coercive system whose solution is exponentially growing or decaying and the rate of growth or decay can be quantified by a (ω,τ*) function in general vector space with a convex functional. Under the same stepsize condition as for strong stability, sharper exponential stability results are derived for explicit and diagonally implicit Runge-Kutta methods and variable coefficients linear multistep methods for nonlinear problems. The new developments in this paper also include their applications to various linear and nonlinear evolution problems. [ABSTRACT FROM AUTHOR]

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform an L² stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector-splitting integrator [C. Lubich and I. V. Oseledets, BIT, 54 (2014), pp. 171-188] when first discretizing the equations and then applying the DLRA. Based on this we propose a projector-splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator [G. Ceruti and C. Lubich, BIT, 62 (2021), pp. 23--44] has more favorable stability properties. Moreover, we explain why the projector-splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented. [ABSTRACT FROM AUTHOR]

We study mathematical models and approximations for the sedimentation in suspensions of rod-like particles. The starting point of our considerations is a coupled system of partial differential equations, previously derived by Helzel and Tzavaras [Multiscale Model. Simul., 15 (2017), pp. 500–536], consisting of a kinetic equation for the rod orientation that is coupled to a macroscopic flow equation. It describes the motion of a suspension of rigid rod-like particles under the influence of gravity. Since the coupled system is high dimensional (five dimensions plus time) we are interested in the derivation of models which can adaptively adjust the level of detail. Here we restrict ourselves to simplified lower dimensional flow situations and derive hierarchies of moment equations which can be interpreted as approximations of the coupled kinetic model. We show that the resulting systems of moment equations are hyperbolic. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mixed system of free boundary problem modeling multilayer tumor growth with time delay. The concept of a time delay goes back to Byrne [Math. Biosci., 144 (1997), pp. 83--117] and the formulation of a multilayer tumor model can be found in Zhou, Escher, and Cui [J. Math. Anal. Appl., 337 (2008), pp. 443--457] and the references therein. In a quasi-steady state approximation of the multilayer tumor growth model with a time delay, He, Xing, and Hu [Math. Methods Appl. Sci., to appear] established a critical value \mu \ast for which the system is linearly stable ifand linearly unstable i under nonflat perturbations. We consider the full system without quasi-steady state simplification in this paper. This is a free boundary system of mixed elliptic, parabolic, and hyperbolic equations. A key difference to our earlier work is the consideration of the full parabolic equation for the nutrient in the model, which makes the system more complex and more challenging. By applying series expansion and Laplace transform methods, we found a threshold 0 such that the unique flat stationary solution is linearly stable if \mu < \mu \ast and linearly unstable if \mu > \mu \ast under nonflat perturbations. This is accomplished with the inverse Laplace transform and a careful study of the spectrum of the system. Some of our results are presented in a more useful abstract form, which would be applicable to other systems. As far as we know, this is the first paper for the study of stability for multilayer tumor models with a time delay and with a parabolic equation for the nutrient. [ABSTRACT FROM AUTHOR]

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H os--R\'enyi and small-world graphs. Applications of the second order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems. [ABSTRACT FROM AUTHOR]

In this article, we extend the time-domain decomposition method described by Lagnese and Leugering [Systems Control Lett., 48 (2003), pp. 229-242] to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coefficients. We provide the design of the iterative method applied to the global first-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a first interpretation of the role of the parameters involved in the iterative process. [ABSTRACT FROM AUTHOR]

Ambiguity sets of probability distributions are used to hedge against uncertainty about the true probabilities of uncertain inputs and random quantities of interest (QoIs). When available, these ambiguity sets are constructed from both data (collected at the initial time and along the boundaries of the physical domain) and concentration-of-measure results on the Wasserstein metric. To propagate the ambiguity sets into the future, we use a physics-dependent equation governing the evolution of cumulative distribution functions (CDFs) obtained through the method of distributions. We investigate the spatio-temporal evolution of data-driven ambiguity sets and their associated guarantees when the random QoIs they describe obey hyperbolic partial differential equations with random inputs. For general nonlinear hyperbolic equations with smooth solutions, the CDF equation is used to propagate the upper and lower envelopes of pointwise ambiguity bands. For linear dynamics, the CDF equation allows us to construct an evolution equation for tighter ambiguity balls. We demonstrate that, in both cases, the ambiguity sets are guaranteed to contain the true (unknown) distribution within a prescribed confidence. [ABSTRACT FROM AUTHOR]

We present a shape-oriented data assimilation strategy suitable for front-tracking tumor growth problems. A general hyperbolic/elliptic tumor growth model is presented as well as the available observations corresponding to the location of the tumor front over time extracted from medical imaging as MRI or CT scans. We provide sufficient conditions allowing one to design a state observer by proving the convergence of the observer model to the target solution for exact parameters. In particular, the similarity measure chosen to compare observations and simulation of tumor contour is presented. A specific joint state-parameter correction with a Luenberger observer correcting the state and a reduced-order Kalman filter correcting the parameters is introduced and studied. We then illustrate and assess our proposed observer method with synthetic problems. Our numerical trials show that state estimation is very effective with the proposed Luenberger observer, but specific strategies are needed to accurately perform parameter estimation in a clinical context. We then propose strategies to deal with the fact that data is very sparse in time and that the initial distribution of the proliferation rate is unknown. The results on synthetic data are very promising, and work is ongoing to apply our strategy on clinical cases. [ABSTRACT FROM AUTHOR]

We rigorously show the nonequilibrium-difusion limit of the compressible Euler-P1 approximation model arising in radiation hydrodynamics as the Mach number tends to zero when the initial data is well prepared. In particular, the efect of the large temperature variation upon the limit is taken into account. The model leads to a singular problem which fails to fall into the category of the classical theory of singular limits for quasi-linear hyperbolic equations. By introducing an appropriate normed space of solutions and exploiting the structure of the system, we establish the uniform local existence of smooth solutions and the convergence of the model to the incompressible nonhomogeneous Euler system coupled with a difusion equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-posedness problem of an initial boundary value problem of a nonlinear wave equation in a cornered space domain with a free boundary. The corner singularity is the essential difficulty and there is no result available even for the linear problem without the symmetry assumptions, i.e., \even" or \odd" traces vanish on the solid boundary, which allows extension from the cornered space domain to the half-space domain, as in the previous works. In this paper, we first obtain an existence result for the initial boundary value problem of linear hyperbolic equations of second order in a cornered space domain without such symmetry assumptions. The key idea is based on the construction of a new auxilliary problem, which allows us to reduce the linear problem to a new one that can be even extended to a half-space domain such that the existence of H² η-solutions can be established. However, due to the lack of the symmetry assumptions, the low regularity of the extended coefficients block us from obtaining the higher regularity of the solutions in the extended domain, which is necessary for the iteration to the nonlinear problem. In order to deal with it, new hyperbolic type and elliptic type estimates in the cornered space domain are established carefully. The results on the general linear problems can be applied to the linearized problem that we are concerned with in this paper. Due to the loss of regularity in the estimates of the linearized problem, a modified Nash-Moser iteration is developed. [ABSTRACT FROM AUTHOR]

In this paper we consider the Runge-Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge- Kutta time-marching is used. By the aid of the equivalent evolution representation with temporal differences of stage solutions, we make a detailed investigation on the matrix transferring process about the energy equations and then present a sufficient condition to ensure the L²-norm stability under the standard Courant-Friedrichs-Lewy condition. If the source term is equal to zero, we achieve the strong (boundedness) stability without the matrix transferring process to multiple-steps time-marching of the RKDG method. By carefully introducing the reference functions and their projections, we obtain the optimal (or suboptimal) error estimate under a mild smoothness assumption on the exact solution, which is independent of the stage number of the RKDG method. Some numerical experiments are also given to verify our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for scalar hyperbolic conservation laws in multidimensions. Compared with previous work for linear hyperbolic equations [W. Guo and Y. Cheng, SIAM J. Sci. Comput., 38 (2016), pp. A3381-A3409, W. Guo and Y. Cheng, SIAM J. Sci. Comput., 39 (2017), pp. A2962-A2992], a class of interpolatory multiwavelets are applied to efficiently compute the nonlinear integrals over elements and edges in DG schemes. The resulting algorithm, therefore, can achieve similar computational complexity as the sparse grid DG method for smooth solutions. Theoretical and numerical studies are performed taking into consideration the accuracy and stability with regard to the choice of the interpolatory multiwavelets. Artificial viscosity is added to capture the shock and only acts on the leaf elements taking advantage of the multiresolution representation. Adaptivity is realized by auto error thresholding based on hierarchical surplus. Accuracy and robustness are demonstrated by several numerical tests. [ABSTRACT FROM AUTHOR]

High-order discontinuous Galerkin (DG) methods promise to be an excellent discretization paradigm for hyperbolic differential equation solvers running on supercomputers, since they combine high arithmetic intensity with localized data access, since they straightforwardly translate into nonoverlapping domain decomposition, and since they facilitate dynamic adaptivity without the need for conformal meshes. An efficient parallel evaluation of DG weak formulation in an MPI+X setting, however, remains nontrivial as dependency graphs over dynamically adaptive meshes change with each mesh refinement or coarsening, as resolution transitions yield nontrivial data flow dependencies, and as data sent along domain boundaries through message passing (MPI) have to be triggered in the correct order. Domain decomposition with MPI alone starts to become insufficient if the mesh changes very frequently, if mesh changes cannot be predicted, and if limiters and nonlinear per-cell solves yield unpredictable costs per cell. We introduce enclave tasking as a task invocation technique for shared memory and MPI+X: It does not assemble any task graph; instead the mesh traversal spawns ready tasks directly. A marker-and-cell approach ensures that tasks feeding into MPI or triggering mesh modifications as well as latency-sensitive or bandwidth-demanding tasks are processed with high priority. The remaining cell tasks form enclaves, i.e., groups of tasks that can be processed in the background. Enclave tasking introduces high concurrency which is homogeneously distributed over the mesh traversal, it mixes memory-intensive volumetric DG calculations with compute-bound Riemann solves, and it helps to overlap communication with computations. Our work focuses on ADER-DG and patch-based finite volumes. Yet, we discuss how the paradigm can be generalized to the whole DG family and finite volume stand-alone solvers. [ABSTRACT FROM AUTHOR]

This paper presents a multilevel convergence framework for multigrid-reduction-intime (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time-stepping operator, analyzed directly and bounded in the norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time-stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge--Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge--Kutta schemes. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hybrid parabolic and hyperbolic equation model, in which a reaction-diffusion equation governs the random movement and settlement of dispersal individuals, while a first-order hyperbolic equation describes the growth of stationary individuals with age structure. We prove the existence and uniqueness of the solution of the model using the monotone method based on a comparison principle. We study the population persistence criteria in terms of four related measures. We numerically investigate how the interplay between population dispersal, reproduction, settlement, and habitat boundary affects the population persistence. [ABSTRACT FROM AUTHOR]

We study the problem of unique recovery of a nonsmooth one-form\scrA and a scalar function q from the Dirichlet to Neumann map,\Lambda\scrA,q, of a hyperbolic equation on a Riemannian manifold (M, g). We prove uniqueness of the one-form\scrA up to the natural gauge, under weak regularity conditions on\scrA, q and under the assumption that (M, g) is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function q. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier integral operator to nonsmooth parameters and functions. [ABSTRACT FROM AUTHOR]

The Escalator Boxcar Train (EBT) numerical method has been designed and widely used by theoretical biologists to compute solutions of one-dimensional structured population models of McKendrick--von Foerster type. Recently the method has been derived for an age-structured twosex population model (Fredrickson--Hoppenstaedt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. The convergence of the EBT method for the Fredrickson--Hoppenstaedt model has not been analyzed, and relevant numerical examples are still missing. In this paper, we derive a simplified EBT method for the "two-sex model" and prove its convergence. However, due to the interest in tracking specified cohorts of individuals, the analytical results cannot be analyzed in the L1 norm. Instead, we embedded the problem in the space of nonnegative Radon measures equipped with the bounded Lipschitz distance (the flat metric). We also present numerical examples to illustrate the results, compute the error in bounded Lipschitz distance, and compare it against the total variation (TV) distance. [ABSTRACT FROM AUTHOR]

Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations with explicit time-stepping schemes, such as strong stability-preserving Runge--Kutta, suffer from time-step restrictions that are significantly worse than what a simple Courant--Friedrichs--Lewy argument requires. In particular, the maximum stable time-step scales inversely with the highest degree in the DG polynomial approximation space and becomes progressively smaller with each added spatial dimension. In this work we introduce a novel approach that we have dubbed the regionally implicit discontinuous Galerkin (RIDG) method to overcome these small time-step restrictions. The RIDG method is based on an extension of the Lax--Wendroff DG method, which previously had been shown to be equivalent to a predictor-corrector approach, where the predictor is a locally implicit spacetime method (i.e., the predictor is something like a block-Jacobi update for a fully implicit spacetime DG method). The corrector is an explicit method that uses the spacetime reconstructed solution from the predictor step. In this work we modify the predictor to include not just local information, but also neighboring information. With this modification we show that the stability is greatly enhanced; in particular, we show that we are able to remove the polynomial degree dependence of the maximum time-step and show how this extends to multiple spatial dimensions. A semianalytic von Neumann analysis is presented to theoretically justify the stability claims. Convergence and efficiency studies for linear and nonlinear problems in multiple dimensions are accomplished using a MATLAB code that can be freely downloaded. [ABSTRACT FROM AUTHOR]

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval (0, T) that depends on the boundary traces of the solution. If the time horizon T is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for T → ∞ the approximation error converges to zero in the sense of the norm in L2(0, 1) with the rate 1/T, if the time interval (0, T) is transformed to the fixed interval (0, 1). Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If T is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations. [ABSTRACT FROM AUTHOR]

The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely, we consider a general class of junction distribution controls and inflow controls and we establish the compactness in L¹ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges. [ABSTRACT FROM AUTHOR]

In this paper, we analyze standard weighted essentially nonoscillatory (WENO) reconstructions and multilevel WENO reconstructions with adaptive order (WENO-AO) using both WENO-JS and WENO-Z weighting. We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The old WENO-AO reconstruction drops to a fixed, base level of approximation when there are discontinuities in the solution, but the new one maintains the accuracy of the largest stencil over which the solution is smooth. Our analysis in the discontinuous case requires that the smoothness indicators do not approach zero as the grid is refined. We provide a condition to ensure this result, but we also show an example where this can fail to occur. That is, we show that WENO reconstructions can fail to maintain the order of approximation of the smallest stencil over which the solution is smooth. We also present numerical results confirming the convergence theory of the old and new WENO-AO reconstructions and compare their performance in solving conservation laws. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of boundary layer solutions for a viscous hyperbolic system transformed via a Cole-Hopf transformation from a singular chemotactic system modeling the initiation of tumor angiogenesis proposed in [H. A. Levine, B. Sleeman, and M. Nilsen-Hamilton, Math. Biosci., 168 (2000), pp. 71-115]. It was previously shown in [Q. Hou, Z. Wang, and K. Zhao, J. Differential Equations, 261 (2016), pp. 5035-5070] that when prescribed with Dirichlet boundary conditions, the system possesses boundary layers at the boundaries in an bounded interval (0, 1) as the chemical diffusion rate (denoted by ε > 0) is small. This paper proceeds to prove the stability of boundary layer solutions and identify the precise structure of boundary layer solutions. Roughly speaking, we justify that the solution with ε > 0 converges to the solution with ε = 0 (outer layer solution) plus the inner layer solution with the optimal rate at order of O(ε½) as ε → 0, where the outer and inner layer solutions are well determined and the relation between outer and inner layer solutions can be explicitly identified. Finally we transfer the results to the original pretransformed chemotaxis system and discuss the implications of our results. [ABSTRACT FROM AUTHOR]

We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori L² error estimates of order k + 1 are obtained for the semidiscrete scheme when piecewise polynomials of degree at most k (k ≥ 0) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We present a new multiorder Monte Carlo algorithm for computing the statistics of stochastic quantities of interest described by linear hyperbolic problems with stochastic parameters. The method is a nonintrusive technique based on a recently proposed high-order energy-based discontinuous Galerkin method for the second-order acoustic and elastic wave equations. The algorithm is built upon a hierarchy of degrees of polynomial basis functions rather than a mesh hierarchy used in multilevel Monte Carlo. Through complexity theorems and numerical experiments, we show that the proposed multiorder method is a valid alternative to the current multilevel Monte Carlo method for hyperbolic problems. Moreover, in addition to the convenience of working with a fixed mesh, which is desirable in many real applications with complex geometries, the multiorder method is particularly beneficial in reducing errors due to numerical dispersion in long-distance propagation of waves. The numerical examples verify that the multiorder approach is faster than the mesh-based multilevel approach for waves that traverse long distances. [ABSTRACT FROM AUTHOR]

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior, which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation lose their efficiency, and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable of capturing the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Strong stability preserving (SSP) methods are designed primarily for time integration of nonlinear hyperbolic PDEs, for which the permissible SSP step size varies from one step to the next. We develop the first SSP linear multistep methods (of order two and three) with variable step size, and prove their optimality, stability, and convergence. The choice of step size for multistep SSP methods is an interesting problem because the allowable step size depends on the SSP coefficient, which in turn depends on the chosen step sizes. The description of the methods includes an optimal step-size strategy. We prove sharp upper bounds on the allowable step size for explicit SSP linear multistep methods and show the existence of methods with arbitrarily high order of accuracy. The effectiveness of the methods is demonstrated through numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we develop a novel two-stage fourth order time-accurate discretization for time-dependent flow problems, particularly for hyperbolic conservation laws. Different from the classical Runge-Kutta (R-K) temporal discretization for first order Riemann solvers as building blocks, the current approach is solely associated with Lax-Wendroff (L-W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using the well-developed four-stage procedure for R-K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L-W type schemes for the construction of a two-stage fourth order scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a shock capturing discontinuous Galerkin method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [A. Hiltebrand and S. Mishra, Numer. Math., 126 (2014), pp. 103-151], our approach is new in that we do not use streamline diffusion stabilization. It is proved that an artificial viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued solution. The result is valid for general systems and for the arbitrary order discontinuous Galerkin method. [ABSTRACT FROM AUTHOR]

We establish the existence result for a global weak solution (respectively, Young measure solution) in the Orlicz-Sobolev space setting for the nonlinear hyperbolic initial boundary value problem to utt = div(σ(Du)) + µ(Δu)t (respectively, utt = div(σ(Du))), where the function σ = @W=@F is continuous and the stored-energy function W : Md×n → R may be nonconvex. Our study is motivated by one-dimensional elastodynamics. The present paper gives first solvability results for nonlinear hyperbolic partial differential equations with nonpower-growth-type nonlinearity for Du in the monotonicity case and in the case with lack of monotonicity. The results are new even for the one-dimensional case with σ(Τ) = lnq(1 + |Τ|)|Τ|p-2Τ + aΤ for q > 0 and p ≥ 2 and a ∈ R; here a > 0 corresponds to the strong ellipticity of W, a = 0 -- the convexity of W, and a < 0 -- the nonconvexity of W under the Andrews-Ball inequality. [ABSTRACT FROM AUTHOR]

This article addresses linear hyperbolic partial differential equations with nonsmooth coefficients and distributional data. Solutions are studied in the framework of Colombeau algebras of generalized functions. Its aim is to prove upper and lower bounds for the singular support of generalized solutions for wave equations with discontinuous coefficients. New existence results with weaker assumptions on the representing families are required and proved. The program is carried through for various types of one- and multidimensional wave equations and hyperbolic systems. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to dissipative wave equations involving two noncommuting self-adjoint operators in a Hilbert space. The main result is that the abstract diffusion phenomenon takes place. Thus solutions of such equations approach solutions of diffusion equations at large times. When the diffusion semigroup has the Markov property and satisfies a Nash-type inequality, we obtain precise estimates for the consecutive diffusion approximations and remainders. We present several important applications including sharp decay estimates for dissipative hyperbolic equations with variable coefficients on an exterior domain. In the nonlocal case we obtain the first decay estimates for nonlocal wave equations with damping; the decay rates are sharp. [ABSTRACT FROM AUTHOR]

Higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions. On the other hand, higher order methods have been applied to hyperbolic problems with less success, as higher order approximations of discontinuous solutions suffer from the Gibbs phenomenon. We extend past work and show that spectral methods yield spectral convergence of moments, even when applied to problems with discontinuous solutions. Besides spectral Fourier methods for periodic domains we also prove high order convergence for adjoint-consistent numerical methods, exemplified by the discontinuous Galerkin finite element method. [ABSTRACT FROM AUTHOR]

This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k + 1)th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k + 1)th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k+2)th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp. [ABSTRACT FROM AUTHOR]

High-order temporal discretizations for hyperbolic conservation laws have historically been formulated as either a method of lines (MOL) or a Lax-Wendroff method. In the MOL viewpoint, the partial differential equation is treated as a large system of ordinary differential equations (ODEs), where an ODE tailored time integrator is applied. In contrast, Lax-Wendroff discretizations immediately convert Taylor series in time to discrete spatial derivatives. In this work, we propose the Picard integral formulation (PIF), which is based on the method of modified fluxes, and is used to derive new Taylor and Runge-Kutta (RK) methods. In particular, we construct a new class of conservative finite difference methods by applying weighted essentially nonoscillatory (WENO) reconstructions to the so-called time-averaged fluxes. Our schemes are automatically conservative under any modification of the fluxes, which is attributed to the fact that classical WENO reconstructions conserve mass when coupled with forward Euler time steps. The proposed Lax-Wendroff discretization is constructed by taking Taylor series of the flux function as opposed to Taylor series of the conserved variables. The RK discretization differs from classical MOL formulations because we apply WENO reconstructions to time-averaged fluxes rather than taking linear combinations of spatial derivatives of the flux. In both cases, we only need one projection onto the characteristic variables per time step. The PIF is generic and lends itself to a multitude of options for further investigation. At present, we present two canonical examples: one based on Taylor, and the other based on the classical RK method. Stability analyses are presented for each method. The proposed schemes are applied to hyperbolic conservation laws in one and two dimensions and the results are in good agreement with current state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel algorithm for computing surface uniformization for surfaces with arbitrary topology. According to the celebrated uniformization theorem, all Riemann surfaces can be classified as elliptic, parabolic, or hyperbolic. Our algorithm is able to work on all these cases by first constructing an initial map onto an appropriate domain, such as a sphere, or a polygon in the plane ℝ² or the hyperbolic disk D, and then morphing the diffeomorphism based on the discrete Beltrami flow algorithm. For high genus surfaces, both the final mapping and the target domain are unknown, which presents a challenge in general. Each such surface can be conformally mapped onto D modulo a discrete subgroup of all fractional linear transforms on D. A conformal copy of the surface, which is also its uniformization domain, can be visualized as a fundamental polygon in D corresponding to the discrete subgroup, where the generators of the discrete subgroup map each side of the fundamental polygon to its corresponding side, giving the conformality information of the surface. The novelty in our method lies in the iterative change of these generators as the diffeomorphism is morphing, which indicates a change of geometry of the target domain to match the geometry of the original surface. Numerical results are presented to show the efficiency and accuracy (in terms of distortion) of our methods as well as to make a comparison to other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

We consider the optimal control of initial-boundary value problems for entropy solutions of scalar hyperbolic conservation laws. In particular, we consider initial-boundary value problems where the initial and boundary data switch between different C1-functions at certain switching points and both the functions and the switching points are controlled. We show that the control-tostate mapping is differentiable in a certain generalized sense, which implies Fréchet-differentiability with respect to the control functions and the switching points for the composition with a tracking type functional, even in the presence of shocks. We also present an adjoint-based formula for the gradient of the reduced objective functional. [ABSTRACT FROM AUTHOR]

The inverse problem of estimating dielectric constants of explosives using boundary measurements of one component of the scattered electric field is addressed. It is formulated as a coefficient inverse problem for a hyperbolic differential equation. After applying the Laplace transform, a new cost functional is constructed and a variational problem is formulated. The key feature of this functional is the presence of the Carleman weight function for the Laplacian. The strict convexity of this functional on a bounded set in a Hilbert space of an arbitrary size is proven. This allows for establishing the global convergence of the gradient descent method. Some results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

A pathway-based mean-field theory (PBMFT) that incorporated the most recent quantitatively measured signaling pathway was recently proposed for the E. coli chemotaxis in [G. Si, T. Wu, Q. Quyang, and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101]. In this paper, we formally derive a new kinetic system of PBMFT under the assumption that the methylation level is locally concentrated, whose turning operator takes into account the dynamical intracellular pathway and hence is more physically relevant. We recover the PBMFT proposed by Si et al. as the hyperbolic limit and connect to the Keller-Segel equation as the parabolic limit of this new model. We also present the numerical evidence to show the quantitative agreement of the kinetic system with the individual based E. coli chemotaxis simulator. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show how a weakly dispersive perturbation of the inviscid Burgers equation improves (enlarges) the space of resolution of the local Cauchy problem. More generally we will review several problems arising from weak dispersive perturbations of nonlinear hyperbolic equations or systems. [ABSTRACT FROM AUTHOR]

A general theorem is proved stating necessary and sufficient conditions for the convergence of the distributions of sums of a random number of independent identically distributed random variables to one-parameter variance-mean mixtures of normal laws. As a corollary, necessary and sufficient conditions for convergence of the distributions of sums of a random number of independent identically distributed random variables to generalized hyperbolic laws are obtained. Convergence rate estimates are presented for a particular case of special continuous time random walks generated by compound doubly stochastic Poisson processes. [ABSTRACT FROM AUTHOR]

The multiclass Lighthill-Whitham-Richards (MCLWR) traffic model, which distinguishes N classes of drivers differing in preferential velocity, gives rise to a system of N strongly coupled, nonlinear first-order conservation laws for the car densities as a function of distance and time. The corresponding velocities involve a hindrance function that depends on the local total density of cars. Since the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic-wise numerical schemes for the MCLWR model become involved. Alternative simple schemes for this model directly utilize that the velocity functions are nonnegative and strictly decreasing, which allows one to construct a new class of schemes by splitting the system of conservation laws into two different first-order quasi-linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps. The new schemes are called Lagrangian-remap (LR) schemes. One version of LR schemes incorporates recent antidiffusive techniques for transport equations. The corresponding subclass of LR schemes are called Lagrangian-antidiffusive-remap (L-AR) schemes. Alternatively, the remap step can be handled by Glimm-like random sampling, which gives rise to a statistically conservative Lagrangian-random-sampling (L-RS) scheme that is less diffusive than other remap techniques. The LR schemes for the MCLWR model are supported by a partial analysis of the L-AR schemes for N = 1, which are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution, and by numerical examples for both L-AR and L-RS subclasses of schemes. [ABSTRACT FROM AUTHOR]

With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many applications one is also interested in the reconstruction of the Fourier coefficients from samples in the spatial domain. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses. In addition, we suggest approaches for determining suitable rank-1 lattices using a component-by-component algorithm. We present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100. [ABSTRACT FROM AUTHOR]

We propose a unified discontinuous Petrov--Galerkin (DPG) framework with optimal test functions for Friedrichs-like systems, which embrace a large class of elliptic, parabolic, and hyperbolic partial differential equations (PDEs). The well-posedness, i.e., existence, uniqueness, and stability, of the DPG solution is established on a single abstract DPG formulation, and two abstract DPG methods corresponding to two different, but equivalent, norms are devised. We then apply the single DPG framework to several linear(ized) PDEs including, but not limited to, scalar transport, Laplace, diffusion, convection-diffusion, convection-diffusion-reaction, linear(ized) continuum mechanics (e.g., linear(ized) elasticity, a version of linearized Navier--Stokes equations, etc.), time-domain acoustics, and a version of the Maxwell's equations. The results show that we not only recover several existing DPG methods, but also discover new DPG methods for both PDEs currently considered in the DPG community and new ones. As a direct consequence of the single abstract DPG framework, all of the resulting DPG methods are shown to be trivially well-posed. We show that the inf-sup constant of the abstract DPG equation is independent of the mesh and is the same order as that of the PDE counterpart. [ABSTRACT FROM AUTHOR]

We consider the second order Cauchy problem ᵋᵘ"ᵋ + |A1/2ᵘᵋ2ᵧAᵘᵋ + ᵘ'ᵋ = 0, ᵘᵋ(0) = ᵘ0 ≠ 0, ᵘ'ᵋ(0) = ᵘ1 where ᵋ > 0, H is a Hilbert space, A is a self-adjoint positive operator on H with dense domain D(A), (ᵘ0, ᵘ1) ∈ D(A) x D(A1/2), and ᵧ > 0. We accurately study the decay as t goes to infinity of the solutions, which exist for every ᵋ small enough. In particular we obtain a new estimate on ᵘ"ᵋ(t) and we show that (1 + t)1/2 ᵧ Aᵘᵋ(t) → ᵘᵋ, ∞ ≠ 0, (1 + t)1+1/1ᵧ A1/2ᵘ'ᵋ (t) → ʋᵋ, ∞ ≠ 0, as t goes to infinity. Moreover we show that the norm of ᵘᵋ, ∞ and ʋᵋ, ∞ does not depend on the initial data. [ABSTRACT FROM AUTHOR]

In this study we use discontinuous conservation laws to model production networks with finite buffers. A network formulation is characterized by the interaction between several production units, described by discontinuous conservation laws, under the assumption that no parts are lost. Therefore we discuss appropriate Riemann problems and coupling conditions which are derived from a regularized model. To solve the discontinuous problem in an accurate and fast way, we introduce a novel algorithm. Based on different settings, the new method is compared numerically with standard approaches. [ABSTRACT FROM AUTHOR]

In this work, we consider the problem of boundary stabilization for a quasilinear 2 x 2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H² exponential stability of the closedloop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4 x 4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them. [ABSTRACT FROM AUTHOR]

This paper considers the global classical solvability (well-posedness) of a mixed initial-boundary value problem for semilinear hyperbolic systems with nonlinear reaction coupling of Lotka--Volterra type. The reaction nonlinearity is not globally Lipschitz in L² and has Lipschitz properties depending on an L∞-norm bound. The well-posedness problem is reformulated in an abstract setting as a modified Cauchy problem with homogeneous boundary conditions and solved based on the Banach contraction mapping theorem. Extra regularity of the local solutions in Sobolev spaces is shown based on Moser-type inequalities. It is shown that global existence of classical solutions holds if a uniform a priori bound on the L∞-norm of the solution and boundary term exists. [ABSTRACT FROM AUTHOR]

We consider implicit-explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such a regime the system relaxes towards a convection-diffusion equation. The first objective of this paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of this paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by a novel reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis of such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation [S. Boscarino and G. Russo, SIAM J. Sci. Comput., 31 (2009), pp. 1926-1945]. Several numerical examples including neutron transport equations confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

We consider the development of Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. The asymptotic behavior of such systems as the relaxation parameter vanishes is governed by a reduced parabolic system, and it is desirable to have schemes that are able to capture the correct diffusive limit. The hyperbolic part becomes stiff when the system relaxes towards the parabolic equation. For this reason, in previous works part of the hyperbolic terms are treated implicitly [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405-2439], [S. Jin and L. Pareschi, J. Comp. Phys., 161 (2000), pp. 312-330], [G. Naldi and L. Pareschi, SIAM J. Numer. Anal., 37 (2000), pp. 1246-1270]. In particular, in [S. Boscarino, L. Pareschi, and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, submitted] the scheme relaxes to an implicit method for the limit diffusive problem, thus avoiding the Δt oc x2 restriction. It would be very desirable to have IMEX schemes which treat the hyperbolic part explicitly, because this allows the use of well-tested space discretization with no modification of the original system. However, the development of such methods presents the difficulty that the characteristic speeds diverge in the diffusive limit making the hyperbolic part very stiff. In this paper we show how to overcome these difficulties with the introduction of particular conditions on the coefficients of the IMEX R-K schemes such that we can treat the stiff component on the hyperbolic part explicitly without any manipulation of the original system. Moreover, the schemes proposed in this paper guarantee a CFL condition independent of the diffusive parameter where a CFL hyperbolic condition in the stiff regime is chosen. Such schemes are shown to have the correct diffusion limit and several numerical results confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]