基于二阶格式的有限体积保正格式

A Positivity-preserving Finite Volume Scheme Based on Second-order Scheme

从二阶线性格式出发,通过对法向通量进行重构,得到非线性两点通量,获得四面体网格上的单元中心型有限体积保正格式.该格式适用于求解间断和各向异性扩散系数问题.无需假设辅助未知量非负,避免了辅助未知量计算出负时"遇负置零"的人为处理方式;并且证明该格式在每个非线性Picard迭代步具有强保正性,即当源项和边界条件非负时,线性化格式的非平凡解是严格大于零的.数值算例验证该格式具有二阶收敛性且是保正的.

Based on a second-order accurate linear scheme,a normal flux is reconstructed to obtain a nonlinear finite volume scheme with a two-point flux discrete stencil on tetrahedral meshes.It is suitable for discontinuous and anisotropic diffusion coefficient problems,and can be generalized to general polyhedral meshes.It is unnecessary to assume that auxiliary unknowns are non-negative,and avoids artificial processing of"setting negative to be zero"in calculating auxiliary unknowns.Moreover,it is proved that the linearized scheme at each nonlinear iteration step satisfies strong positivity-preserving,i.e.,as the source term and boundary condition are non-negative,non-zero solution of the scheme is strictly greater than zero.Numerical tests verify that the scheme has second-order accuracy and is strong positivity-preserving.