We present a finite difference discretization of the incompressible Navier–Stokes equations in cylindrical coordinates. This currently is, to the authors' knowledge, the only scheme available that is demonstrably capable of conserving mass, momentum and kinetic energy (in the absence of viscosity) on both uniform and non-uniform grids. Simultaneously, we treat the inherent discretization issues that arise due to the presence of the coordinate singularity at the polar axis. We demonstrate the validity of the conservation claims by performing a number of numerical experiments with the proposed scheme, and we show that it is second order accurate in space using the Method of Manufactured Solutions.@en

The mass-conserving level-set (MCLS) method is a hybrid level-set (LS)/volume of fluid (VoF) based, interface capturing algorithm that combines the mass conserving properties of the VoF, with the benefits of having an explicit description of the interface of the LS method. The efficiency of the method is a result of the fact that the LS formulation allows evaluation of the VoF-field and VoF-fluxes without reconstruction of the interface in each cell. We present the extension of the MCLS method from its original formulation for Cartesian quadrilateral control volumes to triangular control volumes for optimal geometrical flexibility. The LS field is discretized using a second order discontinuous Galerkin method. After each time-step, a mass-conserving correction is imposed based on the simultaneously convected VoF field. This convection step is performed with a second-order Eulerian–Lagrangian approach, combined with a ‘clipping’ algorithm to project the advected field from the Lagrangian to the Eulerian grid. The MCLS method is shown to be accurately mass conserving and shows second order convergence for three different test cases.@en

The Time Domain Integral Equation method for electromagnetics is an appealing
computational method for many applications in industry. However, its applicability has long been suffering from instabilities. A rigorous analysis of the variational formulation is imperative to the successful design of stable and robust numerical schemes. In this paper, an established functional framework and stability theorem will be extended to the differentiated version of the electric field integral equations, which can be discretized more efficient and is more often used in engineering literature. The extended stability theorem, combined with efficiency requirements, will give guidelines on the choice of
test and basis functions of the space–time Petrov–Galerkin scheme. A discrete equivalence with the collocation method results in the recommendation to choose the quadratic spline basis function in the standard Marching-on-in-Time scheme. Computational experiments confirm that the quadratic spline basis functions have superior stability characteristics compared to the conventional quadratic Lagrange basis functions in time.
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