This report investigates the general theory and methodology of high resolution numerical schemes for one-dimensional hyperbolic conservation laws. The Universal Formula from which 2-level explicit conservative arbitrary-order numerical methods can be derived is developed. This report also explores the issue of linear stability. A new approach to linear stability analysis is presented. The generalized formulation for TVD methods with stable region of -1 ≤ c ≤ 1 proposed. To demonstrate the theories, some third order and fourth order TVD methods are generated.CI
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