General Source Term Research Articles

The full Keller–Segel model is well-posed on nonsmooth domains**This work is dedicated to Prof Willi Jäger.

In this paper we prove that the full Keller–Segel system, a quasilinear strongly coupled reaction-crossdiffusion system of four parabolic equations, is well-posed in the sense that it always admits an unique local-in-time solution in an adequate function space, provided that the initial values are suitably regular. The proof is done via an abstract solution theorem for nonlocal quasilinear equations by Amann and is carried out for general source terms. It is fundamentally based on recent nontrivial elliptic and parabolic regularity results which hold true even on rather general nonsmooth spatial domains. For space dimensions 2 and 3, this enables us to work in a nonsmooth setting which is not available in classical parabolic systems theory. Apparently, there exists no comparable existence result for the full Keller–Segel system up to now. Due to the large class of possibly nonsmooth domains admitted, we also obtain new results for the ‘standard’ Keller–Segel system consisting of only two equations as a special case.

Endpoint Strichartz estimates for charge transfer Hamiltonians

We prove the optimal endpoint Strichartz estimates for Schrodinger equation with charge transfer potentials and a general source term in $\mathbb{R}^n$ for $n\geq3$. The proof is based on using the projection on the scattering states defined implicitly in Rodnianski, Schlag and Soffer \cite{RSS1}, and asymptotic completeness for such systems. The method works as well in the matrix non-selfadjoint case.

Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term

This paper deals with a higher-order wave equation with general nonlinear dissipation and source term \begin{document}$u''+(-Δ)^mu+g(u')=b|u|^{p-2}u, $ \end{document} which was studied extensively when $m=1, 2$ and the nonlinear dissipative term $g(u')$ is a polynomial, i.e., $g(u')=a|u'|^{q-2}u'$. We obtain the global existence of solutions and show the energy decay estimate when $m≥1$ is a positive integer and the nonlinear dissipative term $g$ does not necessarily have a polynomial grow near the origin.

Stochastic modeling of inhomogeneous ocean waves

Refraction of swell waves in coastal waters can result in fast-scale variations of wave statistics due to wave interference. These variations cannot be resolved by wave models based on the radiative transport equation. More advanced models based on quasi-coherent theory, a generalization of the radiative transfer equation, can be coupled or nested into larger-scale models to resolve such local inhomogeneous effects. However, source terms for quasi-coherent models to account for non-conservative and nonlinear effects are not available, which hampers their operational use. In the present work we revisit the derivation of quasi-coherent theory to consistently include a source term for dissipation associated with depth-induced wave breaking. We demonstrate how general source terms can be incorporated in this class of models and compare model simulations with the new dissipation term to laboratory observations of focusing and breaking waves over a submerged shoal. The results show that a consistent derivation of source terms is essential to accurately capture coherent effects in coastal areas. Specifically, our results show that if coherent effects are ignored in the dissipation term, interference effects are strongly exaggerated. With the development of source terms for quasi-coherent models they can be effectively nested inside or otherwise coupled to larger-scale wave models to efficiently improve operational predictive capability of wave models near the coast.

Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.

On symmetry groups of a 2D nonlinear diffusion equation with source

Symmetry analysis of a 2D nonlinear evolutionary equation with mixed spatial derivative and general source term involving the dependent variable and its spatial derivatives is performed. The source terms for which the equation admits nontrivial Lie symmetries are identified for two different forms of the symmetry operator. In one of these cases, the symmetries do not depend on the form of nonlinearities and in the other case, nonlinearities of power, exponential and trigonometric forms are considered. There are no supplementary nonclassical symmetries for the investigated equation. The results reported here generalize the previous results on the 2D heat equation and the 2D Ricci model.

A simplified approach to evaluating severe accident source term for PWR

For early design of NPPs, no specific severe accident source term evaluation was considered. Some general source terms have been used for some NPPs. In order to implement a best estimate, a special source term evaluation should be implemented for an NPP. Traditional source term evaluation approaches (mechanism approach and parametric approach) have some difficulties associated with their implementation. The traditional approaches are not consistent with cost-benefit assessment. A simplified approach for evaluating severe accident source term for PWR is studied. For the simplified approach, a simplified containment event tree is established. According to representative cases selection, weighted coefficient evaluation, computation of representative source term cases and weighted computation, five containment release categories are established, including containment bypass, containment isolation failure, containment early failure, containment late failure and intact containment.

Arctic Ocean freshwater: How robust are model simulations?

The Arctic freshwater (FW) has been the focus of many modeling studies, due to the potential impact of Arctic FW on the deep water formation in the North Atlantic. A comparison of the hindcasts from ten ocean‐sea ice models shows that the simulation of the Arctic FW budget is quite different in the investigated models. While they agree on the general sink and source terms of the Arctic FW budget, the long‐term means as well as the variability of the FW export vary among models. The best model‐to‐model agreement is found for the interannual and seasonal variability of the solid FW export and the solid FW storage, which also agree well with observations. For the interannual and seasonal variability of the liquid FW export, the agreement among models is better for the Canadian Arctic Archipelago (CAA) than for Fram Strait. The reason for this is that models are more consistent in simulating volume flux anomalies than salinity anomalies and volume‐flux anomalies dominate the liquid FW export variability in the CAA but not in Fram Strait. The seasonal cycle of the liquid FW export generally shows a better agreement among models than the interannual variability, and compared to observations the models capture the seasonality of the liquid FW export rather well. In order to improve future simulations of the Arctic FW budget, the simulation of the salinity field needs to be improved, so that model results on the variability of the liquid FW export and storage become more robust.

Axially symmetric pseudo-Newtonian hydrodynamics code

We develop a numerical hydrodynamics code using a pseudo-Newtonian formulation that uses the weak field approximation for the geometry, and a generalized source term for the Poisson equation that takes into account relativistic effects. The code was designed to treat moderately relativistic systems such as rapidly rotating neutron stars. The hydrodynamic equations are solved using a finite volume method with High Resolution Shock Capturing (HRSC) techniques. We implement several different slope limiters for second order reconstruction schemes and also investigate higher order reconstructions. We use the method of lines (MoL) to convert the mixed spatial-time partial differential equations into ordinary differential equations (ODEs) that depend only on time. These ODEs are solved using 2nd and 3rd order Runge-Kutta methods. The Poisson equation for the gravitational potential is solved with a multigrid method. In order to confirm the validity of our code, we carry out four different tests including one and two dimensional shock tube tests, stationary star tests of both non-rotating and rotating models and radial oscillation mode tests for spherical stars. In the shock tube tests, the code shows good agreement with analytic solutions which include shocks, rarefaction waves and contact discontinuities. The code is found to be stable and accurate: for example, when solving a stationary stellar model the fractional changes in the maximum density, total mass, and total angular momentum per dynamical time are found to be $3 \times 10^{-6}$, $5 \times 10^{-7}$ and $2 \times 10^{-6}$, respectively. We also find that the frequencies of the radial modes obtained by the numerical simulation of the steady state star agree very well with those obtained by linear analysis.

Results of Predictive Fokker–Planck Modelling of NBI Deuterons in ITER

First an analytical formalism is presented for calculating the source distribution of ions generated by neutral beam injection (NBI) in tokamak plasmas. A general NBI ion source term, applicable to studies in the phase space up to 6 dimensions, is provided for neutral beams with finite thickness and divergence. Further, using this source term for the envisaged NBI in ITER, we carry out 3D Fokker–Planck modelling of the steady-state deuteron distribution function of NBI produced fast deuterons relaxing on bulk plasma components. For two basic ITER scenarios we demonstrate the poloidal profiles of the beam deuteron density, of the NBI generated current as well as of the NBI power deposition to bulk electrons and ions. Further, we evaluate the capability of gamma and NPA diagnostics of NBI ions in ITER and demonstrate the sensitivity of the distributions of NBI generated ions to different ITER operation scenarios.

Seismic wave-equation demigration/migration

Reverse-time migration is a well-known method based on a single-scattering approximation; it is designed to obtain seismic images in the case of a complex subsurface. It can, however, be a very time-consuming task because the number of computations is directly proportional to the number of processed sources. In the context of velocity model-building, iterative approaches require that one derives a series of migrated sections for different velocity models. We propose to replace the summation over sources by a summation over depth offsets or time delays defined in the subsurface. For that, we have developed a new relationship between two migrated sections obtained for two different velocity models. Starting from one of the two images, we obtain a second section correctly and efficiently. For each time delay, we compute a generalized source term by extending the concept of exploding reflector to nonzero offset. We obtain the final migrated section by solving the same wave equation in the perturbed model with the modified source term. Our work included testing the methodology on 2D synthetic data sets, particularly when the initial and perturbed velocity models differ greatly.

Source term models for fine particle resuspension from indoor surfaces

Understanding the dispersion of contaminants inside buildings is important for improving indoor air quality (IAQ). Detailed information on the dispersion profile within a room is required to design active protection systems and to develop countermeasure strategies against potential threats from particulate based agents. A number of computational fluid dynamics (CFD) codes in the public and commercial domain can simulate contaminant dispersion inside a building. One of the critical boundary conditions required by these CFD codes is a resuspension source term model. This paper develops general source term models for particle resuspension from indoor surfaces based on dimensional analysis. First, the physical mechanisms responsible for fine particle resuspension from indoor surfaces are investigated and relevant parameters are identified. Then, three different models are developed using dimensional analysis and published resuspension data in the literature. Finally, the models are evaluated against independent experimental data that were not used to determine the model coefficients.

Existence of global weak solutions to a special system of Euler equation with a source (II): General case

In this paper, we apply the maximum principle and the theory of compensated compactness to establish an existence theorem for global weak solutions to the Cauchy problem of the non-strictly hyperbolic system—a special system of Euler equation with a general source term.

An evolutionary double-well problem

We establish the existence theorem and study the long time behaviour of the following PDE problem:(0.1){ut−div∇W(∇u)−f(x)=0in Ω×(0,−∞),∇W(∇u)⋅n|∂Ω×(0,∞)=0,u(x,0)=u0(x)in Ω where W is a specially given quasiconvex double-well function and f∈L2(Ω) is a given function independent of time t. In particular, the existence theorem is established for general given source term f, the long time behaviour is analyzed under the assumption that ∫Ωf(x)dx=0.The system is an evolutionary quasimonotone system. We believe that the existence of solutions established here is stronger than the usual Young Measure solution and is the first of its kind. The existence of a compact ω-limit set as t→∞ is also established under some non-restrictive conditions.

Global Existence and Energy Decay of Solutions to a Petrovsky Equation with General Nonlinear Dissipation and Source Term

Abstract We consider the nonlinearly damped semilinear Petrovsky equation and prove the global existence of its solutions by means of the stable set method in combined with the Faedo–Galerkin procedure. Furthermore, we study the asymptotic behavior of solutions when the nonlinear dissipative term 𝑔 does not necessarily have a polynomial growth near the origin.

On the Fundamental Diagram of Traffic Flow

We present a new fluid-dynamical model of traffic flow. This model generalizes the model of Aw and Rascle [SIAM J. Appl. Math., 60 (2000), pp. 916-938] and Greenberg [SIAM J. Appl. Math., 62 (2001), pp. 729-745] by prescribing a more general source term to the velocity equation. This source term can be physically motivated by experimental data, when taking into account relaxation and reaction time. In particular, the new model has a (linearly) unstable regime as observed in traffic dynamics. We develop a numerical code that solves the corresponding system of balance laws. Applying our code to a wide variety of initial data, we find the observed inverse-$\lambda$ shape of the fundamental diagram of traffic flow.

A noise-controlled free shear flow

The adjoint of the perturbed and linearized compressible viscous flow equations is formulated in such a way that its solution can be used to optimize control actuation in order to reduce flow-generated sound. We apply it to a direct numerical simulation of a randomly excited two-dimensional mixing layer, with inflow vorticity-thickness Reynolds number 500 and free-stream Mach numbers 0.9 and 0.2. The control actuation is implemented as general source terms in the flow equations (body forces, mass sources, and internal energy sources) with compact support near the inflow boundary. The noise to be reduced is defined by a space-time integral of the mean-square pressure fluctuations on a line parallel to the mixing layer in the acoustic field of the low-speed stream. Both the adjoint and flow equations are solved numerically and without modelling approximations. The objective is to study the mechanics of the noise generation and its control. All controls reduce targeted noise with very little required input power, with the most effective (the internal energy control) reducing the noise intensity by 11 dB. Numerical tests confirm that the control is not by any simple acoustic cancellation mechanism but instead results from a genuine change of the flow as a source of sound. The comparison of otherwise identical flows with and without control applied shows little change of the flow's gross features: the evolution and pairings of the energetic structures, turbulence kinetic energy, spreading rate, and so on are superficially unchanged. However, decomposition of the flow into empirical eigenfunctions, as surrogates for Fourier modes in the non-periodic streamwise direction, shows that the turbulence structures advect downstream more uniformly. This change appears to be the key to reducing their acoustic efficiency, a perspective that is clarified by comparing the randomly excited mixing layer to a harmonically excited mixing layer, which is relatively quiet because it is highly ordered. Unfortunately, from the perspective of any practical implementation with actuators, the optimized control identified has a complex spatial and temporal structure, but it can be simplified. Two empirical eigenmodes were required to represent it sufficiently to reduce the targeted noise intensity by about 50%. Optimization of a simple single-degree-of-freedom control with an ad hoc spatial structure is less effective.

Single Grid Error Estimation Using Error Transport Equation

This paper presents an approach to quantify the discretization error as well as other errors related to mesh size using the error transport equation (ETE) technique on a single grid computation. The goal is to develop a generalized algorithm that can be used in conjunction with computational fluid dynamics (CFD) codes to quantify the discretization error in a selected process variable. The focus is on applications where the conservation equations are solved for primitive variables, such as velocity, temperature and concentration, using finite difference and/or finite volume methods. An error transport equation (ETE) is formulated. A generalized source term for the ETE is proposed based on the Taylor series expansion and accessible influence coefficients in the discretized equation. Representative examples, i.e., one-dimensional convection diffusion equation, two-dimensional Poisson equation, two-dimensional convection diffusion equation, and non-linear one-dimensional Burger’s equation are presented to verify this method and elucidate its properties. Discussions are provided to address the significance and possible potential applications of this method to Navier-Stokes solvers.

Source term identification in 1-D IHCP

We propose stable numerical solutions for the simultaneous identification of temperature, temperature gradient, and general source terms in the one-dimensional inverse heat conduction problem (IHCP). The numerical solution consists of a regularization procedure, based on the mollification method,and a marching scheme for the solution of the stabilized problem. The stability, error analysis and implementation of the algorithm are presented together with a set of numerical results.

An integral equation basic to the BGK model for flow in a cylindrical tube

The classical BGK model for describing the flow of a rarefied gas in a cylindrical tube is used as a starting point for developing an equivalent integral-equation formulation of the considered class of problems. While the problems of Poiseuille flow and thermal creep in a cylindrical tube have already been well solved in terms of a pseudo problem obtained from an integral-equation definition of the problems, the development of the relevant integral equation is given here for a larger class of problems. In particular it is noted that a general inhomogeneous source term in the balance equation and a general inhomogeneous term in the boundary condition are both included in the considered model, and, as a special case, the integral-equation formulation for the case of specular reflection at the surface of the tube is also developed.