Before revision: We design well-balanced numerical algorithms with first-, second-, and third-order of accuracy for the spherically symmetry evolution of a compressible fluid  on a Schwarzschild black hole .
After revision: We design well-balanced numerical algorithms with first-, second-, and third-order of accuracy for the spherically symmetry evolution of a compressible fluid on a Schwarzschild curved background, we design a class of well-balanced numerical algorithms with first-order or second-order of accuracy .