Traditional　methods　such　as　the　finite　difference　method,　the　finite　element　method,　and　the　finite　volume　method　are　all　based　on　continuous　interpolation.　In　general,　if　discontinuity　occurred,　the　calculation　result　would　show　low　accuracy　and　poor　stability.　In　this　paper,　the　numerical　manifold　method　is　used　to　capture　numerical　discontinuities,　in　a　one-dimensional　space.　It　is　verified　that　the　high-degree　Legendre　polynomials　can　be　selected　as　the　local　approximation　without　leading　to　linear　dependency,　a　notorious　＂nail＂　issue　in　Numerical　Manifold　Method.　A　series　of　numerical　tests　are　carried　out　to　evaluate　the　performance　of　the　proposed　method,　suggesting　that　the　accuracy　by　the　numerical　manifold　method　is　higher　than　that　by　the　later　finite　difference　method　and　finite　volume　method　using　the　same　number　of　unknowns.