In this paper, we develop third-order conservative sign-preserving and steady-state-preserving time integrations and seek their applications in multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction for the ith species, denoted as zi, 1<=i<=M, should be between 0 and 1, where M is the total number of species. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum-principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the ux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this paper, we will construct third-order conservative sign-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with large time step and preserves the steady-state. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.

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We consider an incompressible viscous flow without surface tension in a finite-depth domain of two dimensions, with free top boundary and fixed bottom boundary. This system is governed by the Navier-Stokes equations in this moving domain and the transport equation on the moving boundary. In this paper, we construct a stable numerical scheme to simulate the evolution of this system by discontinuous Galerkin method, and discuss the error analysis of the fluid under certain assumptions. Our formulation is mainly based on the geometric structure introduced earlier in the literature, and the natural energy estimate, which is rarely used in the numerical study of this system before.

Most biochemical processes involve macromolecules in solution. The corresponding electrostatics is of central importance for understanding their structures and functions. An accurate and efficient numerical scheme is introduced to evaluate the corresponding electrostatic potential and force by solving the governing Poisson-Boltzmann equation. This paper focuses on the following issues: (i) the point charge singularity problem, (ii) the dielectric discontinuity problem across a molecular surface, and (iii) the infinite domain problem. Green's function associated with the point charges plus a harmonic function is introduced as the zeroth order approximation to the solution to solve the point charge singularity problem. A jump condition capturing finite difference scheme is adopted to solve the discontinuity problem across molecule surfaces, where a body-fitting grid is used. The infinite domain problem is solved by mapping the outer infinite domain into a finite domain. The corresponding stiffness matrix is symmetric and positive definite, therefore, fast algorithm such as preconditioned conjugate gradient method can be applied for inner iteration. Finally, the resulting scheme is second order accurate for both the potential and its gradient.

We propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry. By introducing a generalized vorticity-stream formulation, the divergence free constraints are automatically satisfied. In addition, with explicit treatment of the nonlinear terms and local vorticity boundary condition, the Navier–Stokes (MHD, respectively) equation essentially decouples into 2 (4, respectively) scalar equation and thus the scheme is very efficient. Moreover, with proper discretization of the nonlinear terms, the scheme preserves both energy and helicity identities numerically. This is achieved by recasting the nonlinear terms (convection, vorticity stretching, geometric source, Lorentz force and electro-motive force) in terms of Jacobians. This conservative property is valid even in the presence of the pole singularity for axisymmetric flows. The exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity. Numerical examples have demonstrated both accuracy and efficiency of the scheme. Finally, local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost.