Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.

In this paper we deal with the numerical approximation of integro-differential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are high-order accurate for large time regimes. Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic high-order schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.

Abstract. We consider a model for dusty gas flow that consists of the compressible Euler equations for the gas coupled to a similar (but pressureless) system of equations for the mass, momentum, and energy of the dust. These sets of equations are coupled via drag terms and heat transfer. A high-resolution wave-propagation algorithm is used to solve the equations numerically. The one-dimensional algorithm is shown to give agreement with a shock tube test problem in the literature. The two-dimensional algorithm has been applied to model expolsive volcanic eruptions in which an axisymmetric jet of hot dusty gas is injected into the atmosphere and the expected behavior is observed at two different vent velocities. The methodology described here, with extensions to three dimensions and adaptive mesh refinement, is being used for more detailed studies of volcanic jet processes. Key words. Finite volume methods, high-resolution methods, volcanic flows, dusty gas, plumes, jets, shocks

Flux-gradient and source term balancing for certain high resolution shock-capturing schemes

Abstract not found

The shallow water equations are a commonly accepted approximation governing tsunami propagation. Numerically capturing certain features of local tsunami inundation requires solving these equations in their physically relevant conservative form, as integral conservation laws for depth and momentum. This form of the equations presents challenges when trying to numerically model global tsunami propagation, so often the best numerical methods for the local inundation regime are not suitable for the global propagation regime. The different regimes of tsunami flow belong to different spatial scales as well, and require correspondingly different grid resolutions. The long wavelength of deep ocean tsunamis requires a large global scale computing domain, yet near the shore the propagating energy is compressed and focused by bathymetry in unpredictable ways. This can lead to large variations in energy and run-up even over small localized regions. We have developed a finite volume method to deal with the diverse flow regimes of tsunamis. These methods are well suited for the inundation regime—they are robust in the presence of bores and steep gradients, or drying regions, and can capture the inundating shoreline and run-up features. Additionally, these methods are well-balanced, meaning that they can appropriately model global propagation. To deal with the disparate spatial scales, we have used adaptive refinement algorithms originally developed for gas dynamics, where often steep variation is highly localized at a given time, but moves throughout the domain. These algorithms allow evolving Cartesian sub-grids that can move with the propagating waves and highly resolve local inundation of impacted areas in a single global scale computation. Because the dry regions are part of the computing domain, simple rectangular cartesian grids eliminate the need for complex shoreline-fitted mesh generation. Science of Tsunami Hazards, Vol. 24, No. 5, page 319 (2006)

The three-dimensional matlab graphics routines were developed by

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e. it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm. AMS subject classification: Primary 65M99; Secondary 35L65 Key words: Hyperbolic systems of conservation and balance laws, semi-discrete schemes, Saint-Venant system of Shallow Water equations, non-oscillatory reconstructions, channels with irregular geometry. 1 The Shallow-water Model We consider the shallow water equations along channels with non-uniform rectangular cross sections and bottom topography. The model describes flows that are nearly horizontal and can be obtained by averaging the Euler equations over the channel cross section [3], resulting in the balance law ∂A ∂t

Instability of power-law fluid flows down an incline subjected to wind stress J.P. Pascal a, S.J.D. D’Alessio b, *

Numerische Mathematik manuscript No. (will be inserted by the editor)