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The emergence of mutant lineages within a viral species has become a public health problem, as the existing treatments and drugs are usually more effective on the original lineages than in the mutant ones. The following manuscript presents mathematical models that describe the emergence of these lineages. In order to reduce the damage and possible casualties that can be attributed to these more contagious microorganisms, the theory of optimal control is introduced and a more sophisticated model is proposed to reduce the mutant growth compared to the original one. The analytical study of these models allows us to obtain an overview of the expected behavior over time, which is validated with numerical simulations.

In this paper the projection hybrid FV/FE method presented in Busto et al. 2014 is extended to account for species transport equations. Furthermore, turbulent regimes are also considered thanks to the $k-\varepsilon$ model. Regarding the transport diffusion stage new schemes of high order of accuracy are developed. The CVC Kolgan-type scheme and ADER methodology are extended to 3D. The latter is modified in order to profit from the dual mesh employed by the projection algorithm and the derivatives involved in the diffusion term are discretized using a Galerkin approach. The accuracy and stability analysis of the new method are carried out for the advection-diffusion-reaction equation. Within the projection stage the pressure correction is computed by a piecewise linear finite element method. Numerical results are presented, aimed at verifying the formal order of accuracy of the scheme and to assess the performance of the method on several realistic test problems., Comment: arXiv admin note: text overlap with arXiv:1802.10585

We construct, analyse and assess various schemes of second order of accuracy in space and time for model advection-diffusion-reaction differential equations. The constructed schemes are meant to be of practical use in solving industrial problems and are derived following two related approaches, namely ADER and MUSCL-Hancock. Detailed analysis of linear stability and local truncation error are carried out. In addition, the schemes are implemented and assessed for various test problems. Empirical convergence rate studies confirm the theoretically expected accuracy in both space and time.

In this paper a new hybrid semi-implicit finite volume / finite element (FV/FE) scheme is presented for the numerical solution of the compressible Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered meshes in two and three space dimensions. The chosen grid arrangement consists of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based / face-based staggered dual mesh. The governing equations are discretized in conservation form. The nonlinear convective terms of the equations, as well as the viscous stress tensor and the heat flux, are discretized on the dual mesh at the aid of an explicit local ADER finite volume scheme, while the implicit pressure terms are discretized at the aid of a continuous $\mathbb{P}^{1}$ finite element method on the nodes of the primal mesh. In the zero Mach number limit, the new scheme automatically reduces to the hybrid FV/FE approach forwarded in \cite{BFTVC17} for the incompressible Navier-Stokes equations. As such, the method is asymptotically consistent with the incompressible limit of the governing equations and can therefore be applied to flows at all Mach numbers. Due to the chosen semi-implicit discretization, the CFL restriction on the time step is only based on the magnitude of the flow velocity and not on the sound speed, hence the method is computationally efficient at low Mach numbers. In the chosen discretization, the only unknown is the scalar pressure field at the new time step. Furthermore, the resulting pressure system is symmetric and positive definite and can therefore be very efficiently solved with a matrix-free conjugate gradient method. In order to assess the capabilities of the new scheme, we show computational results for a large set of benchmark problems that range from the quasi incompressible low Mach number regime to compressible flows with shock waves.

The purpose of this work is to introduce a novel POD-Galerkin strategy for the hybrid finite volume/finite element solver introduced in Berm\'udez et al. 2014 and Busto et al. 2018. The interest is into the incompressible Navier-Stokes equations coupled with an additional transport equation. The full order model employed in this article makes use of staggered meshes. This feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes. The reduced order model presented herein accounts for velocity, pressure, and a transport-related variable. The pressure term at both the full order and the reduced order level is reconstructed making use of a projection method. More precisely, a Poisson equation for pressure is considered within the reduced order model. Results are verified against three-dimensional manufactured test cases. Moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed.

In this article we present a novel staggered semi-implicit hybrid finite-volume/finite-element (FV/FE) method for the resolution of weakly compressible flows in two and three space dimensions. The pressure-based methodology introduced in Berm\'udez et al. 2014 and Busto et al. 2018 for viscous incompressible flows is extended here to solve the compressible Navier-Stokes equations. Instead of considering the classical system including the energy conservation equation, we replace it by the pressure evolution equation written in non-conservative form. To ease the discretization of complex spatial domains, face-type unstructured staggered meshes are considered. A projection method allows the decoupling of the computation of the density and linear momentum variables from the pressure. Then, an explicit finite volume scheme is used for the resolution of the transport diffusion equations on the dual mesh, whereas the pressure system is solved implicitly by using continuous finite elements defined on the primal simplex mesh. Consequently, the CFL stability condition depends only on the flow velocity, avoiding the severe time restrictions that might be imposed by the sound velocity in the weakly compressible regime. High order of accuracy in space and time of the transport diffusion stage is attained using a local ADER (LADER) methodology. Moreover, also the CVC Kolgan-type second order in space and first order in time scheme is considered. To prevent spurious oscillations in the presence of shocks, an ENO-based reconstruction, the minmod limiter or the Barth-Jespersen limiter are employed. To show the validity and robustness of our novel staggered semi-implicit hybrid FV/FE scheme, several benchmarks are analysed, showing a good agreement with available exact solutions and numerical reference data from low Mach numbers, up to Mach numbers of the order of unity.