An adaptive E–scheme for degenerate, viscous balance laws is presented. Taking into account natural diffusion, numerical viscosity is locally reduced to a minimum. Numerical experiments demonstrate the improved accuracy of the adaptive scheme. Explicit and implicit three–point E–schemes are monotone, TVD and nonlinearly stable. A high–resolution version of the adaptive E–scheme is derived and tested in experiments. The latter is not necessarily monotone, but TVD.

An adaptive E-scheme for possibly degenerate, viscous conservation laws is presented. The scheme makes use of both given and numerical diffusion to establish the E-property. In the degenerate case it reduces to local Lax–Friedrichs. Both explicit and time-implicit E-schemes are monotone and TVD. Numerical experiments demonstrate the robustness and improved accuracy of the adaptive scheme.

Fluorescence loss in photobleaching (FLIP) is a modern microscopy method for visualization of transport processes in living cells. Although FLIP is widespread, an automated reliable analysis of image data is still lacking. This paper presents a framework for modeling and simulation of FLIP sequences as reaction–diffusion systems on segmented cell images. The cell geometry is extracted from microscopy images using the Chan–Vese active contours algorithm (IEEE Trans Image Process 10(2):266–277, 2001). The PDE model is subsequently solved by the automated Finite Element software package FEniCS (Logg et al. in Automated solution of differential equations by the finite element method. Springer, Heidelberg, 2012). The flexibility of FEniCS allows for spatially resolved reaction diffusion coefficients in two (or more) spatial dimensions.

Background Fluorescence loss in photobleaching (FLIP) is a widely used imaging technique, which provides information about protein dynamics in various cellular regions. In FLIP, a small cellular region is repeatedly illuminated by an intense laser pulse, while images are taken with reduced laser power with a time lag between the bleaches. Despite its popularity, tools are lacking for quantitative analysis of FLIP experiments. Typically, the user defines regions of interest (ROIs) for further analysis which is subjective and does not allow for comparing different cells and experimental settings. Results We present two complementary methods to detect and quantify protein transport and aggregation in living cells from FLIP image series. In the first approach, a stretched exponential (StrExp) function is fitted to fluorescence loss (FL) inside and outside the bleached region. We show by reaction–diffusion simulations, that the StrExp function can describe both, binding/barrier–limited and diffusion-limited FL kinetics. By pixel-wise regression of that function to FL kinetics of enhanced green fluorescent protein (eGFP), we determined in a user-unbiased manner from which cellular regions eGFP can be replenished in the bleached area. Spatial variation in the parameters calculated from the StrExp function allow for detecting diffusion barriers for eGFP in the nucleus and cytoplasm of living cells. Polyglutamine (polyQ) disease proteins like mutant huntingtin (mtHtt) can form large aggregates called inclusion bodies (IB’s). The second method combines single particle tracking with multi-compartment modelling of FL kinetics in moving IB’s to determine exchange rates of eGFP-tagged mtHtt protein (eGFP-mtHtt) between aggregates and the cytoplasm. This method is self-calibrating since it relates the FL inside and outside the bleached regions. It makes it therefore possible to compare release kinetics of eGFP-mtHtt between different cells and experiments. Conclusions We present two complementary methods for quantitative analysis of FLIP experiments in living cells. They provide spatial maps of exchange dynamics and absolute binding parameters of fluorescent molecules to moving intracellular entities, respectively. Our methods should be of great value for quantitative studies of intracellular transport.

A novel concept for calibrating depositional models is presented. In this approach, transport coefficients are determined from well output measurements. Finite element implementation of the multi-lithology models and their duals is automated by the DOLFIN Python interface.

Nonlinear systems of time dependent partial differential equations of real life phenomena tend to be very complicated to solve numerically. One common approach to solve such problems is by applying operator splitting which introduces subproblems that are easier to handle. The solutions of the subsystems are usually glued together by either the Godunov method (first-order) or the Strang method (second-order). However, the accuracy of such an approach may be very hard to analyse because of the complexity of the equations involved. The purpose of the present paper is to introduce another line of reasoning concerning the accuracy of operator splitting. Let us assume that a fully coupled and implicit discretization of the complete system has been developed. Under appropriate conditions on the continuous problem, such discretizations provide reasonable and convergent approximations. As in the continuous case, operator splitting can be utilized to obtain tractable algebraic subsystems. The problem we address in the present paper is to obtain a bound on the difference between the fully coupled implicit discrete solutions, u h, and the solutions, u h, s, obtained by applying operator splitting to these discrete equations. Suppose we know that u h converges to the analytical solution u as the grid is properly refined and, applying the results from this paper, that u h, s converges toward u h under grid refinement. Then, by the triangle inequality, also the splitting approximation u h, s converges toward the analytical solution u and convergence is thus obtained for an approximation that, from a practical point of view, is easier to compute.

A non-oscillatory, high resolution reconstruction method on quadrilateral meshes in two dimensions (2D) is presented. It is a two-dimensional extension of Marquina's hyperbolic method. The generalization to quadrilateral meshes allows the method to simulate realistic flow problems in complex domains. An essential point in the construction of the method is a second order accurate approximation of gradients on an irregular, quadrilateral mesh. The resulting scheme is optimal in the sense that it is third order accurate and the reconstruction requires only nearest neighbour information. Numerical experiments are presented and the computational results are compared to experimental data.

The Camassa-Holm equation is a conservation law with a non-local flux that models shallow water waves and features soliton solutions with a corner at their crests, so-called peakons. In the present paper a finite-volume method is developed to simulate the dynamics of peakons. This conservative scheme is adaptive, high resolution and stable without any explicit introduction of artificial viscosity. A numerical simulation indicates that a certain plateau shaped travelling wave solution breaks up in time.

A third order conservative reconstruction, in the context of finite volume schemes for hyperbolic conservation laws, is constructed based on logarithmic functions. This logarithmic method reconstructs without the use of a limiter, any preprocessing of input data, special treatments for local extrema, or shock solutions. Also the method is local in the sense that data from only the nearest neighbors are required.We test the new reconstruction method in several numerical experiments, including nonlinear systems in one and two space dimensions.

A class of high-order reconstruction methods based on logarithmic functions is presented. Inspired by Marquina's hyperbolic method, we introduce a double logarithmic ansatz of fifth order of accuracy. Low variation is guaranteed by the ansatz and (slope-) limiting is avoided. The method can reconstruct smooth extrema without order reduction. Fifth order of convergence is verified in a numerical experiment governed by the nonlinear Euler system. Numerical experiments, including the Osher-Shu shock/acoustic interaction, are presented.

In this paper, a high-resolution finite volume method for calculating solutions to a hyperbolic conservation law is presented. The method works in two space dimensions on general domains and uses curvilinear meshes. A non-trivial estimation of gradients needed for the reconstructions is presented. The paper contains examples of numerical solutions of the Enter gas dynamics equations. Copyright (c) 2005 John Wiley & Sons, Ltd.

Relaxed, essentially non-oscillating schemes for nonlinear conservation laws are presented. Exploiting the relaxation approximation, it is possible to avoid the nonlinear Riemann problem, characteristic decompositions, and staggered grids. Nevertheless, convergence rates up to fourth order are observed numerically. Furthermore, a relaxed, piecewise hyperbolic scheme with artificial compression is constructed. Third order accuracy of this method is proved. Numerical results for two-dimensional Riemann problems in gas dynamics are presented. Finally, the relation to central schemes is discussed.

Numerical schemes based on relaxation are typically central difference schemes. In the case of supersonic flows, however, central differences are unphysical approximations. Introducing a shift in the relaxation approximation relaxed upwind schemes are constructed. Similar as central relaxed schemes, the new upwind versions also avoid the nonlinear Riemann problem and staggered grids. In addition, they simulate the physical domain of dependence correctly even in transonic flow regimes. The performance of the methods is illustrated by an acoustic shock interaction in gas dynamics.

The present paper reviews a particular argument how to obtain error estimates for Godunov-type schemes applied to conservation laws with source terms. In the case of stiff sources-so-called relaxation problems-a special monotonicity property of the source term is required. Several examples satisfying this requirement are presented. However, this paper is not a complete review on the topic. The reader is referred to the contribution by Tadmor at the same conference, the lecture notes (Tadmor, 1998) and the proceedings (Aregba-Driollet and Natalini, 1999).

A system arising in the modeling of oil-recovery processes is analyzed. It consists of a hyperbolic conservation law governing the saturation and an elliptic equation for the pressure. By an operator splitting approach, an approximate solution is constructed. For this approximation appropriate a-priori bounds are derived. Applying the Arzela-Ascoli theorem, local existence and uniqueness of a classical solution for the original hyperbolic-elliptic system is proved. Furthermore, convergence of the approximation generated by operator splitting towards the unique solution follows. It is also proved that the unique solution is stable with respect to perturbations of the initial data.

Well-posedness of a constant-coefficient, first-order, hyperbolic system is equivalent to the existence of a symmetrizer. This symmetrizer is usually constructed in Fourier space and generally depends on the wave number. Nevertheless, many physical systems have constant symmetrizers which define a quadratic entropy. This paper presents physically relevant systems with relaxation which are stiffly well-posed, but no constant symmetrizer and no quadratic, strictly convex entropy exists.

The Cauchy problem for linear constant-coefficient hyperbolic systems ut+∑jA(j)uxj=(1/δ)Bu+Cu in d space dimensions is analyzed. Here (1/δ)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non-trivial null-space. A concept of stiff well-posedness is introduced that ensures solution estimates independent of 0<δ⪡1. Stiff well-posedness is characterized algebraically and—under mild assumptions on B—is shown to be equivalent to the existence of a limit of the L2-solution as δ→0. The evolution of the limit is governed by a reduced hyperbolic system, the so-called equilibrium system, which is related to the original system by a phase speed condition. We also show that stiff well-posedness—which is a weaker requirement than the existence of an entropy—leads to the validity of an asymptotic expansion. As an application, we consider a linearized version of a generic model of two-phase flow in a porous medium and show stiff well-posedness using a general result on strictly hyperbolic systems. To confirm the theory, the leading terms of the asymptotic expansion are computed and compared with a numerical solution of the full problem.

The Cauchy problem for linear constant{coeecient hyperbolic systems u t + P j A (j) u x j = (1==)Bu + Cu in d space dimensions is analyzed. Here (1==)Bu is a large relaxation term, and we are mostly interested in the critical case where B has a non{trivial null{space. A concept of stii well{ posedness is introduced that ensures solution estimates independent of 0 < 1. Under suitable assumptions, an asymptotic expansion for small is presented. Furthermore, we give sharp conditions for the existence of a limit of the L 2 {solution as tends to zero. The evolution of this limit is governed by a reduced hyperbolic system, the so{called equilibrium system. The original system and the equilibrium system are related by a phase speed condition. The theory is applied to a linearized version of a generic model of two{ phase ow in a porous medium. Stii well{posedness of this system is analyzed in detail, and the leading terms of the expansion are computed numerically.

A straightforward semi-implicit finite-difference method approximating a system of conservation laws including a stiff relaxation term is analyzed. We show that the error, measured in $\Lone$, is bounded by $O(\sqrt{\dt})$ independent of the stiffness, where the time step $\dt$ represents the mesh size. As a simple corollary we obtain that solutions of the stiff system converge toward the solution of an equilibrium model at a rate of $O(\delta^{1/3})$ in $\Lone$ as the relaxation time $\delta$ tends to zero.