In　this　paper,　we　present　a　unified　approach　to　study　superconvergence　behavior　of　the　local　discontinuous　Galerkin　(LDG)　method　for　high-order　time-dependent　partial　differential　equations.　We　select　the　third　and　fourth　order　equations　as　our　models　to　demonstrate　this　approach　and　the　main　idea.　Superconvergence　results　for　the　solution　itself　and　the　auxiliary　variables　are　established.　To　be　more　precise,　we　first　prove　that,　for　any　polynomial　of　degree　k,　the　errors　of　numerical　fluxes　at　nodes　and　for　the　cell　averages　are　superconvergent　under　some　suitable　initial　discretization,　with　an　order　of　.　We　then　prove　that　the　LDG　solution　is　-th　order　superconvergent　towards　a　particular　projection　of　the　exact　solution　and　the　auxiliary　variables.　As　byproducts,　we　obtain　a　-th　and　-th　order　superconvergence　rate　for　the　derivative　and　function　value　approximation　separately　at　a　class　of　Radau　points.　Moreover,　for　the　auxiliary　variables,　we,　for　the　first　time,　prove　that　the　convergence　rate　of　the　derivative　error　at　the　interior　Radau　points　can　reach　as　high　as　.　Numerical　experiments　demonstrate　that　most　of　our　error　estimates　are　optimal,　i.e.,　the　error　bounds　are　sharp.