A Conservative High-precision Preserving Scheme For Compressible Fluid Mechanics Equations And Radiation Transport Equations

Radiative hydrodynamics problems commonly exist in many fields such as laser fusion(ICF),weapon physics,and astrophysics.One of the indispensable and important approaches to solve them is numerical simulation.Radiative hy-drodynamics is usually described by the compressible hydrodynamic equations and the radiative transfer equation.The above mentioned equations have some important physical properties,such as conservation and preservation of posi-tivity.Conservation means that the mass,momentum and total energy of the system remain unchanged.Positivity-preserving property refers to that physical quantities such as density,internal energy and radiative intensity should always be positive or non-negative.The physical properties satisfied by the physical equations should also hold in the corresponding numerical methods,which is an important manifestation of the robustness of numerical methods.Howev-er,these properties tend to be lost in the numerical simulation,in particular for high order approximations and non-conservative schemes.Therefore,it is of great theoretical and practical values to study the high order conservative and positivity-preserving numerical methods of these two types of equations.In the field of laser fusion,there are many three-dimensional cylinder symmetry models,such as sphere-shape capsules with fusion fuel and cylinder-shape hohlraum for laser indirect-drive model.We usually describe and simulate numerically these models with the Lagrangian framework in the cylindrical coordinates.Hence,symmetry-preservation is another important issue for the calculation of such problems.With the above research background,the content of this paper includes two parts.In the first part,we mainly study the second order positivity-preserving and symmetry-preserving conservative Lagrangian schemes for compressible flows in the cylindrical coordinates.Taking the compressible Euler equations as an ex-ample and using the two-state Riemann solver,we ensure the positivity of density and internal energy by controlling the change rate of the control volume.For the case in two-dimensional cylindrical coordinates,we consider the properties of positivity-preserving and symmetry-preserving simultaneously.To do that,we establish a local polar coordinate system inside each cell.Polynomial reconstruc-tion and limitation procedure are performed within this local polar coordinates.Besides,the pressure in the source term is computed by a special technique.With these,we finally get a second order numerical scheme,which can maintain pos-itivity and symmetry,satisfy physical conservation and geometric conservation law.Our numerical tests also demonstrate these good properties.In the second part,we focus on the positivity-preserving discontinuous Galerkin(DG)methods for radiative transfer equations.The DG method has many ad-vantages such as high order accuracy,geometric flexibility,suitability for h-and p-adaptivity,local conservation and high parallel efficiency.Therefore,it is wide-ly used in the numerical solutions of various hyperbolic equations,including the radiative transfer equations.In this part,we first take the linear hyperbolic equa-tions as an example to design a conservative positivity-preserving DG method.By means of the discrete ordinate method(DOM),we can apply it directly to solve radiative transfer equations.In the one-dimensional case,we prove a key-result that the DG solver based on the polynomial space Pk for any k can main-tain the positivity of the cell average when the inflow boundary value and the source term are both positive,therefore the positivity-preserving limiter in[X.Zhang&C.-W.Shu,J.Comput.Phys.,229(2010)8919-8934]can be used to get the conservative positivity-preserving DG method.Unfortunately,in two-dimensions this is no longer the case based on either P1 spaces or Qk spaces.We then propose the idea of augmenting the DG space by adding additional functions to get the new augmented DG spaces denoted as Rk.The DG solver based on Rk spaces can have the accuracy of degree k + 1 and can maintain the positivity of cell averages.The limitation procedure mentioned above can be performed to achieve the conservation and positivity-preserving property.Some numerical ex-amples axe given to illustrate the good performance of our positivity-preserving DG schemes.