An adaptive version of Glimm's scheme

An a posteriori error estimation strategy for the boundary element solution of two-dimensional heat conduction problem is presented. The error estimate formulation is based on the boundary element equations and employs a least squares fit of the solutions obtained from these equations. A local error estimate is evaluated for the unknown quantity in each element. The global error estimates are found from local (element) quantities. The range and accuracy of the proposed strategy is very good as demonstrated by problems containing smooth and non-smooth solutions.

In this paper, we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform L1 boundness for the solutions. The existence theory generalizes the classical Glimm theory for systems, for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax. We construct the solutions by the Glimm scheme through the wave tracing method. One of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves. A new analysis is introduced to verify the consistency of the wave tracing procedure. The entropy functional is used to study the L1 boundedness.

Typical nonlinear wave interaction problems involve strong waves moving through a background of weak disturbance. Previous existence theorems and error analysis for computations are usually restricted to more idealized situations such as small data or single equations. We consider here the problem of a single strong discontinuity interacting with a weak background for general hyperbolic systems of conservation laws. We obtain the stability, consistency theorems and upper bounds of the truncation errors for the Glimm scheme and for a front tracking method. The major error in the Glimm scheme is the error generated by the strong discontinuity. This error is reduced when a front tracking method is applied to follow the location of the strong discontinuity. This demonstrates an advantage of front tracking methods in one‐space dimension.

These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: (1) Meaning of a conservation equation and definition of weak solutions. (2) Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. (3) Shock waves: Rankine–Hugoniot equations and admissibility conditions. (4) Genuinely nonlinear and linearly degenerate characteristic fields. Centered rarefaction waves. The general solution of the Riemann problem. Wave interaction estimates. (5) Weak solutions to the Cauchy problem, with initial data having small total variation. Approximations generated by the front-tracking method and by the Glimm scheme. (6) Continuous dependence of solutions w.r.t. the initial data, in the L 1 distance. (7) Characterization of solutions which are limits of front tracking approximations. Uniqueness of entropy-admissible weak solutions. (8) Vanishing viscosity approximations. (9) Extensions and open problems. The survey is concluded with an Appendix, reviewing some basic analytical tools used in the previous sections.Throughout the exposition, technical details are mostly left out. The main goal of these notes is to convey basic ideas, also with the aid of a large number of figures.KeywordsWeak SolutionCauchy ProblemHyperbolic SystemRiemann ProblemPiecewise Constant FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.

We consider weakly coupled systems of nonlinear hyperbolic conservation laws on moving surfaces. As in the Euclidean space, see, for example, (Levy, Commun Partial Differ Equ 17(3–4):657–698, 1992, [9], Rohde, Weakly coupled systems of hyperbolic conservation laws. PhD thesis, Mathematische Fakultat der Albert-Ludwigs-Universitat Freiburg 1996, [16]), the coupling is realized by a source term, which only depends on (x, t) and the unknown function u(x, t) but not on its derivatives. Scalar conservation laws on moving surfaces were considered in Dziuk et al., Interfaces Free Boundaries 15:202–236, 2013, [4]), Lengeler and Muller (J Differ Equ 254(4):1705–1727, 2013, [10]). The velocity of the surface is given by a smooth function, and we assume the surface to be compact. We prove the existence for an entropy solution. First, we consider the regularized parabolic problem with viscosity parameter \(\varepsilon \) and show that there exists a weak solution by decoupling and linearizing the problem. Then, we prove the boundedness of this solution in \(L^\infty (G_T)\), use standard regularity results to prove that this solution is a solution in the classical sense, and show uniform boundedness in \(L^\infty (G_T)\) and \(W^{1,1}(G_T)\) with respect to \(\varepsilon \).

This contribution is devoted to fully discrete discontinuous Galerkin approximations of systems of hyperbolic conservation laws in one space dimension. Its focus is on a posteriori error estimators which are obtained by a combination of a reconstruction approach with the relative entropy stability framework. It was shown in earlier works that for certain numerical fluxes, the error estimators are of the same order as the true error before shock formation. For discontinuous solutions, the use of the relative entropy methodology prevents convergence of the error estimator. We investigate whether a part of the error estimator (related to residuals) is convergent post-shock and whether it is useful as an error indicator or a smoothness indicator.

This paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions. The main assumption is that the system under study admits a convex entropy extension. It is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interior solutions are generated by Godunov schemes, any Lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values. This in particular gives a trace formula for the flux functions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws. It is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general. The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. For example, we give example which show that Godunov schemes produce numerical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleinik entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of × quadratic system with a uniformly characteristic boundary condition.

By means of the 2-stage ROW methods, applied to a stiff model, Stetter's global error estimation is studied in the smooth phase. It turns out that this error estimation, which is based on theh→0 asymptotic, remains also reliable forh≫0 if the parameters of the method are chosen in a certain way. The relations between the effectiveness of the error estimation, the stiffness of the problem and the parameters of the method are investigated by means of an error expansion, the coefficients of which are bounded for decreasingh as well as for increasing stiffness.

We present a class of high‐order weighted essentially nonoscillatory (WENO) reconstructions based on relaxation approximation of hyperbolic systems of conservation laws. The main advantage of combining the WENO schemes with relaxation approximation is the fact that the presented schemes avoid solution of the Riemann problems due to the relaxation approach and high‐resolution is obtained by applying the WENO approach. The emphasis is on a fifth‐order scheme and its performance for solving a wide class of systems of conservation laws. To show the effectiveness of these methods, we present numerical results for different test problems on multidimensional hyperbolic systems of conservation laws. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Collections of linear or nonlinear operator equations Au = f are considered which may represent (i) differential or integral equations or (ii) finite-dimensional approximations. Input sets of coefficients a or data f are admitted. The envelope of the set of solutions is to be constructed where this boundary refers (i) to the ränge of values of the solutions or (ii) to a finite-dimensional space. The construction employs either topological boundary mapping or truncated Taylor expansions. Estimates of the local procedural errors are due to suitable a priori sets and interval mathematics. The relation between local and global error estimates is due to boundary mapping or an auxiliary inverse-monotone operator B. The operator B is constructed for the case of arbitrary linear ordinary differential equations with boundary or initial conditions, provided the admitted A satisfy a mild condition.KeywordsBoundary MappingOperator EquationRemainder TermDiscrete AnalogyOuter ApproximationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

High dynamic aircraft is a very attractive new generation vehicles, in which provides near space aviation with large flight envelope both speed and altitude, for example the hypersonic vehicles. The complex flight environments for high dynamic vehicles require high accuracy and stability navigation scheme. Since the conventional Strapdown Inertial Navigation System (SINS) and Global Position System (GPS) federal integrated scheme based on EKF (Extended Kalman Filter) is invalidation in GPS single blackout situation because of high speed flight, a new high precision and stability integrated navigation approach is presented in this paper, in which the SINS, GPS and Celestial Navigation System (CNS) is combined as a federal information fusion configuration based on nonlinear Unscented Kalman Filter (UKF) algorithm. Firstly, the new integrated system state error is modeled. According to this error model, the SINS system is used as the navigation solution mathematic platform. The SINS combine with GPS constitute one error estimation filter subsystem based on UKF to obtain local optimal estimation, and the SINS combine with CNS constitute another error estimation subsystem. A non-reset federated configuration filter based on partial information is proposed to fuse two local optimal estimations to get global optimal error estimation, and the global optimal estimation is used to correct the SINS navigation solution. The χ 2 fault detection method is used to detect the subsystem fault, and the fault subsystem is isolation through fault interval to protect system away from the divergence. The integrated system takes advantages of SINS, GPS and CNS to an immense improvement for high accuracy and reliably high dynamic navigation application. Simulation result shows that federated fusion of using GPS and CNS to revise SINS solution is reasonable and availably with good estimation performance, which are satisfied with the demands of high dynamic flight navigation. The UKF is superior than EKF based integrated scheme, in which has smaller estimation error and quickly convergence rate.

In recent papers the technique for a local and global error estimation and the local-global step size control were presented to solve both ordinary differential equations and semi-explicit index 1 differential-algebraic systems by multistep methods with any reasonable accuracy attained automatically. Now those results are extended to the concept of multistep extrapolation, and the paper demonstrates with numerical examples how such methods work in practice. Especially, we develop an efficient technique to calculate higher derivatives of a numerical solution with Hermite interpolating polynomials. The necessary theory is also provided.

This paper proposes a new methodology for robotic offline programming (OLP) addressing the issue of automatic program generation directly from 3D CAD models and verification through online 3D reconstruction. Limitations of current OLP include manufacturing tolerances between CAD and workpieces and inaccuracies in workpiece placement and modelled work cell. These issues are addressed and demonstrated through surface scanning, registration, and global and local error estimation. The method allows the robot to adjust the welding path designed from the CAD model to the actual workpiece. Alternatively, for non-repetitive tasks and where a CAD model is not available, it is possible to interactively define the path online over the scanned surface.KeywordsRoot Mean Square ErrorControl PointRobot TrajectoryWelding PathVisibility ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present an overview on some recent works in collaboration with S. Bianchini (see Bianchini and Modena in Lagrangian representation for solution to general systems of conservation laws [9] and the Ph.D. thesis Modena in Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for hyperbolic systems of conservation laws [15]), in which we propose a way to describe BV solutions to hyperbolic systems of conservation laws in one space dimension from a Lagrangian point of view.