  A class of high-order shock-capturing schemes, P$_n$T$_m$-BVD (Deng et al.,
J. Comp. Phys., 386:323-349, 2019; Comput. & Fluids, 200:104433, 2020.)
schemes, have been devised to solve the Euler equations with substantially
reduced numerical dissipation, which enable high-resolution simulations to
resolve flow structures of wider range scales. In such simulations with low
dissipation, errors of round-off level might grow and contaminate the numerical
solutions. A typical example of such problems is the loss of symmetry in the
numerical solutions for physical problems of symmetric configurations even if
the schemes are mathematically in line with the symmetry rules. In this study,
the mechanisms of symmetry-breaking in a finite volume framework with the
P$_4$T$_2$-BVD reconstruction scheme are thoroughly examined. Particular
attention has been paid to remove the possible causes due to the lack of
associativity in floating-point arithmetic which is associated with round-off
errors. Modifications and new techniques are proposed to completely remove the
possible causes for symmetry breaking in different components of the
P$_4$T$_2$-BVD finite volume solver. Benchmark tests that have symmetric
solution structures are used to verify the proposed methods. The numerical
results demonstrate the perfect symmetric solution structures.
