Semi-implicit schemes for compressible fluids in elastic pipes and a-posteriori sub-cell finite volume limiting techniques for semi-implicit Discontinuous Galerkin schemes for hyperbolic conservation laws on staggered meshes

In the present work we solve systems of partial differential equations (PDE) for hyperbolic conservation laws using semi-implicit numerical methods on staggered meshes applied both to the class of finite volume (FV) and both to the family of high order Discontinuous Galerkin (DG) finite elements schemes. In particular, we want to show that these new semi-implicit schemes can be applied in several fields of applied sciences, such as in geophysical flows and compressible fluids in compliant tubes. Inside this thesis we distinguish two big parts. First, we consider staggered semi-implicit schemes for compressible viscous fluids flowing in elastic pipes. This topic is very important in several practical applications of civil, environmental, industrial and biomedical engineering. Here, we analyse the accuracy and the computational efficiency of fully explicit and semi-implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric rigid and elastic pipes. We consider two families of differential models that can be used to predict the pressure and velocity distribution along the tube. One is the so called 2Dxr PDE model which is derived from the full compressible Navier-Stokes equations under the assumptions of a hydrostatic pressure and an axially symmetric geometry. The second family is a simple 1D non-conservative PDE system based on the cross-sectionally averaged version of the Navier-Stokes equations in cylindrical coordinates. In this last case, the use of a simple steady friction model is not enough to simulate the wall friction phenomena in highly transient regime. As a consequence the wall friction model has to be frequency dependent and, following previous studies present in the literature, we consider the classes of convolution integral (CI) models and instantaneous acceleration (IA) models. We carry out a rather complete analysis of the previously-mentioned methods for the simulation of flows characterized by fast transient regime in rigid and compliant tubes. The numerical results show that the convolution integral models are clearly better than instantaneous acceleration models concerning accuracy. Moreover, for CI models, instead of computing the convolution integrals, which is very time- and memoryconsuming, we express these methods via a set of additional ODEs for appropriate auxiliary variables. This trick improves the computational efficiency of these methods substantially, since it avoids the direct computation of the convolution integral. In addition, semi-implicit finite volume methods are significantly superior to classical explicit finite volume schemes in terms of computational efficiency, however, providing the same level of accuracy. We then proceed by extending the finite volume discretization of the 1D and 2Dxr PDE models to arbitrary high-order of accuracy in space introducing a new SIDG scheme on staggered meshes. Both models include the effects of the viscosity and of the wall motion. The nonlinear convective terms are discretized explicitly by using a classical RKDG scheme of arbitrary high-order of accuracy in space and third order of accuracy in time. The continuity equation is integrated over the elements that belong to the main grid, while the momentum equation is integrated over the control volumes of the edgebased staggered dual grid. Inserting the discrete momentum equation into the discrete continuity leads to a mildly nonlinear algebraic system for the degrees of freedom of the pressure, which is solved by using the (nested) Newton method of Brugnano, Casulli and Zanolli. We use the -method in order to get second order of accuracy in time for the implicit part of the scheme. In addition, the schemes have to obey only a mild CFL condition based on the fluid velocity and not based on the sound speed; consequently these schemes work also in the low Mach number regime and even in the incompressible limit of the Navier-Stokes equations. This is a very important property, which is the so-called asymptotic preserving (AP) property of the scheme. We carry out several numerical tests in order to validate this novel family of numerical methods against available exact solutions and experimental data. We also report numerical convergence tables in order to show that the new schemes indeed achieve high order of accuracy in space. In the second part of the thesis, we present a new class of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin schemes on staggered Cartesian grids for the solution of the 1D and 2D shallow water equations (SWE) and of the Euler equations both expressed in conservative form. Here, the starting point is the unlimited arbitrary high order accurate staggered SIDG scheme proposed by Dumbser and Casulli (2013). For this metho d, the mass conservation equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. According to the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. Furthermore, the staggered SIDG is also extended to the Euler equations of compressible gasdynamics. Here, the governing PDE are rewritten using a flux vector splitting technique. The convective terms are updated using an explicit Runge-Kutta DG integrator. Then, the discrete momentum equation, which is integrated again on the dual grid, is coupled with the discrete energy equation that is discretized on the control volumes of the main grid. The pressure is efficiently obtained solving a linear system combined with an iterative Picard iteration procedure.