D. R. van der Heul

A conservative pressure-correction method for flow at all speeds

Abstract A new fully conservative Mach-uniform staggered scheme is discussed. With this scheme one can compute flow with a Mach number ranging from the incompressible limit M ↓0 up to supersonic flow M >1, with nearly uniform efficiency and accuracy. Earlier methods are based on a nonconservative discretisation of the energy equation. This results in small discrepancies in the computed shock speed for the Euler equations. The new method has similar Mach-uniform properties as the earlier methods, but is found to converge to the correct weak solution.

Design of Temporal Basis Functions for Time Domain Integral Equation Methods With Predefined Accuracy and Smoothness

A key parameter in the design of integral equation methods for transient electromagnetic scattering is the definition of temporal basis functions. The choice of temporal basis functions has a profound impact on the efficiency and accuracy of the numerical scheme. This paper presents a framework for the design of temporal basis functions with predefined accuracy and varying smoothness properties. The well-known shifted Lagrange basis functions naturally fit in this framework. New spline basis functions will be derived that have the same interpolation accuracy as shifted Lagrange basis functions and with the added advantage of being smooth. Numerical experiments show the positive influence of smoothness on the quadrature error in the numerical integration procedure. The global accuracy in time of the numerical scheme based on shifted Lagrange and spline basis functions has been experimentally analyzed. For a given interpolation error the experiments confirm the expected accuracy for the shifted Lagrange basis functions, but remarkably show a higher order of accuracy for the spline basis functions.

Uniformly Effective Numerical Methods for Hyperbolic Systems

Abstract A unified method to compute compressible and incompressible flows is presented. Accuracy and efficiency do not degrade as the Mach number tends to zero. A staggered scheme solved with a pressure correction method is used. The equation of state is arbitrary. A Riemann problem for the barotropic Euler equations with nonconvex equation of state is solved exactly and numericaly. A hydrodynamic flow with cavitation in which the Mach number varies between 10−3 and 20 is computed. Unified methods for compressible and incompressible flows are further discussed for the flow of a perfect gas. The staggered scheme with pressure correction is found to have Mach-uniform accuracy and efficiency, and for the fully compressible case the accuracy is comparable with that of established schemes for compressible flows.

The accuracy of temporal basis functions used in the TDIE method

A key parameter in the design of integral equation methods for transient electromagnetic scattering is the choice of temporal basis functions. Newly constructed basis functions have to meet requirements on accuracy, smoothness and efficiency, while the requirement of bandlimitedness is dropped for the nonlinear case. An analysis of the interpolation accuracy will justify the widespread use of the shifted Lagrange basis functions, because these have optimal accuracy, but introduce nonsmoothness in the calculated fields. Alternatively, a novel spline basis function is proposed that has optimal accuracy under an additional smoothness constraint. Computational results confirm the expected smoothness and accuracy.

A Staggered Scheme for Hyperbolic Conservation Laws Applied to Computation of Flow with Cavitation

We demonstrate the advantages of discretizing on a staggered grid for the computation of solutions to hyperbolic systems of conservation laws arising from instationary flow of an inviscid fluid with an arbitrary equation of state. The method is used to compute unsteady sheet cavitation, with the homogeneous equilibrium model for two phase flow. Due to strong variations in the speed of sound almost incompressible as well as highly compressible regions occur simultaneously. The compressible pressure correction solution method is able to handle both, because accuracy and efficiency are uniform in the Mach number.

Two-Phase Flow Solved by High Order Discontinuous Galerkin Method

In this article we present a discretisation of a one-dimensional, hyperbolic model for two-phase pipe flow based on a Discontinuous Galerkin Finite Element Method with a viscous regularisation to suppress the Gibbs phenomenon.

A Unified Method for Compressible and Incompressible Flows with General Equation of State

A unified method for computing incompressible and compressible flows is presented. The method is an extension of a staggered scheme with pressure correction for incompressible flows. The method is extended to fluids with arbitrary equations of state, and compared with the Osher scheme for Riemann problems. An application is shown to a hydrodynamic flow with cavitation, in which the Mach number varies between 10-3 in the water and 20 in the transition zone between water and vapor.