Supersonic flows over both simple and complex geometries involve features over a wide spectrum of spatial and temporal scales, whose resolution in a numerical solution is of significant importance for accurate predictions in engineering applications. While CFD has been greatly developed in the last 30 years, the desire and necessity to perform more complex, high fidelity simulations still remains. The present thesis has introduced two major innovations regarding the fidelity of numerical solutions of the compressible \ns equations. The first one is the development of new a priori mesh quality measures for the Finite Volume (FV) method on mixed-type (quadrilateral/triangular) element meshes. Elementary types of mesh distortion were identified expressing grid distortion in terms of stretching, skewness, shearing and non-alignment of the mesh. Through a rigorous truncation error analysis, novel grid quality measures were derived by emphasizing on the direct relation between mesh distortion and the quality indicators. They were applied over several meshes and their ability was observed to identify faithfully irregularly-shaped small or large distortions in any direction. It was concluded that accuracy degradation occurs even for small mesh distortions and especially at mixed-type element mesh interfaces the formal order of the FV method is degraded no matter of the mesh geometry and local mesh size. Therefore, in the present work, the high-order Discontinuous Galerkin (DG) discretization of the compressible flow equations was adopted as a means of achieving and attaining high resolution of flow features on irregular mixed-type meshes for flows with strong moving shocks. During the course of the thesis a code was developed and named HoAc (standing for High Order Accuracy), which can perform via the domain decomposition method parallel $p$-adaptive computations for flows with strong shocks on mixed-type element meshes over arbitrary geometries at a predefined arbitrary order of accuracy. In HoAc in contrast to other DG developments, all the numerical operations are performed in the computational space, for all element types. This choice constitutes the key element for the ability to perform $p$-adaptive computations along with modal hierarchical basis for the solution expansion. The time marching of the DG discretized Navier-Stokes system is performed with the aid of explicit Runge-Kutta methods or with a matrix-free implicit approach. The second innovation of the present thesis, which is also based on the choice of implementing the DG method on the regular computational space, is the development of a new $p$-adaptive limiting procedure for shock capturing of the implemented DG discretization. The new limiting approach along with positivity preserving limiters is suitable for computations of high speed flows with strong shocks around complex geometries. The unified approach for $p$-adaptive limiting on mixed-type meshes is achieved by applying the limiters on the transformed canonical elements, and it is fully automated without the need of ad hoc specification of parameters as it has been done with standard limiting approaches and in the artificial dissipation method for shock capturing. Verification and validation studies have been performed, which prove the correctness of the implemented discretization method in cases where the linear elements are adequate for the tessellation of the computational domain both for subsonic and supersonic flows. At present HoAc can handle only linear elements since most grid generators do not provide meshes with curved elements. Furthermore, p-adaptive computations with the implemented DG method were performed for a number of standard test cases for shock capturing schemes to illustrate the outstanding performance of the proposed $p$-adaptive limiting approach. The obtained results are in excellent agreement with analytical solutions and with experimental data, proving the excellent efficiency of the developed shock capturing method for the DG discretization of the equations of gas dynamics. / -

Parciální diferenciální rovnice hrají důležitou v inženýrských aplikacích. Často je možné tyto rovnice řešit pouze přibližně, tj. numericky. Z toho důvodu vzniklo množství diskretizačních metod pro řešení těchto rovnic. Uvedená nespojitá Galerkinova metoda se zdá jako velmi obecná metoda pro řešení těchto rovnic, především pak pro hyperbolické systémy. Naším cílem je řešit úlohy aeroakustiky, přičemž šíření akustických vln je popsáno pomocí linearizovaných Eulerových rovnic. A jelikož se jedná o hyperbolický systém, byla vybrána právě nespojitá Galerkinova metoda. Mezi nejdůležitější aspekty této metody patří schopnost pracovat s geometricky složitými oblastmi, možnost dosáhnout metody vysokého řádu a dále lokální charakter toho schématu umožnuje efektivní paralelizaci výpočtu. Nejprve uvedeme nespojitou Galerkinovu metodu v obecném pojetí pro jedno- a dvoudimenzionalní úlohy. Algoritmus následně otestujeme pro řešení rovnice advekce, která byla zvolena jako modelový případ hyperbolické rovnice. Metoda nakonec bude testována na řadě verifikačních úloh, které byly formulovány pro testování metod pro výpočetní aeroakustiku, včetně oveření okrajových podmínek, které, stejně jako v případě teorie proudění tekutin, jsou nedílnou součástí výpočetní aeroakustiky.

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This dissertation is a contribution to the development of a unified model of continuum mechanics, describing both fluids and elastic solids as a general continua, with a simple material parameter choice being the distinction between inviscid or viscous fluid, or elastic solids or visco-elasto-plastic media. Additional physical effects such as surface tension, rate-dependent material failure and fatigue can be, and have been, included in the same formalism. The model extends a hyperelastic formulation of solid mechanics in Eulerian coordinates to fluid flows by means of stiff algebraic relaxation source terms. The governing equations are then solved by means of high order ADER Discontinuous Galerkin and Finite Volume schemes on fixed Cartesian meshes and on moving unstructured polygonal meshes with adaptive connectivity, the latter constructed and moved by means of a in- house Fortran library for the generation of high quality Delaunay and Voronoi meshes. Further, the thesis introduces a new family of exponential-type and semi- analytical time-integration methods for the stiff source terms governing friction and pressure relaxation in Baer-Nunziato compressible multiphase flows, as well as for relaxation in the unified model of continuum mechanics, associated with viscosity and plasticity, and heat conduction effects. Theoretical consideration about the model are also given, from the solution of weak hyperbolicity issues affecting some special cases of the governing equations, to the computation of accurate eigenvalue estimates, to the discussion of the geometrical structure of the equations and involution constraints of curl type, then enforced both via a GLM curl cleaning method, and by means of special involution-preserving discrete differential operators, implemented in a semi-implicit framework. Concerning applications to real-world problems, this thesis includes simulation ranging from low-Mach viscous two-phase flow, to shockwaves in compressible viscous flow on unstructured moving grids, to diffuse interface crack formation in solids.