An Alternating Direction Explicit Method for Time Evolution Equations with Applications to Fractional Differential Equations

We derive and analyze the alternating direction explicit (ADE) method for time evolution equations with the time-dependent Dirichlet boundary condition and with the zero Neumann boundary condition. The original ADE method is an additive operator splitting (AOS) method, which has been developed for treating a wide range of linear and nonlinear time evolution equations with the zero Dirichlet boundary condition. For linear equations, it has been shown to achieve the second order accuracy in time yet is unconditionally stable for an arbitrary time step size. For the boundary conditions considered in this work, we carefully construct the updating formula at grid points near the boundary of the computational domain and show that these formulas maintain the desired accuracy and the property of unconditional stability. We also construct numerical methods based on the ADE scheme for two classes of fractional differential equations. We will give numerical examples to demonstrate the simplicity and the computational efficiency of the method.Comment: 25 pages, 1 figure, 7 table

Some aspects of RANS based jet noise prediction

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

AREPO-RT: Radiation hydrodynamics on a moving mesh

We introduce AREPO-RT, a novel radiation hydrodynamic (RHD) solver for the unstructured moving-mesh code AREPO. Our method solves the moment-based radiative transfer equations using the M1 closure relation. We achieve second order convergence by using a slope limited linear spatial extrapolation and a first order time prediction step to obtain the values of the primitive variables on both sides of the cell interface. A Harten-Lax-Van Leer flux function, suitably modified for moving meshes, is then used to solve the Riemann problem at the interface. The implementation is fully conservative and compatible with the individual timestepping scheme of AREPO. It incorporates atomic Hydrogen (H) and Helium (He) thermochemistry, which is used to couple the ultra-violet (UV) radiation field to the gas. Additionally, infrared radiation is coupled to the gas under the assumption of local thermodynamic equilibrium between the gas and the dust. We successfully apply our code to a large number of test problems, including applications such as the expansion of ${\rm H_{II}}$ regions, radiation pressure driven outflows and the levitation of optically thick layer of gas by trapped IR radiation. The new implementation is suitable for studying various important astrophysical phenomena, such as the effect of radiative feedback in driving galactic scale outflows, radiation driven dusty winds in high redshift quasars, or simulating the reionisation history of the Universe in a self consistent manner.Comment: v2, accepted for publication in MNRAS, changed to a Strang split scheme to achieve second order convergenc

Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

Incompressible lagrangian fluid flow with thermal coupling

A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method