Browsing by Author "Shi, Jian"

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    Arbitrary-order high resolution schemes for model hyperbolic conservation laws
    (1992) Shi, Jian
    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.
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    Arbitrary-order numerical schemes for linear hyperbolic systems
    (1992) Shi, Jian
    This report is an extension of the work carried out in [16]. In [16] we defined arbitrary-order numerical methods for model scalar hyperbolic equation. In this report we extended these methods to linear hyperbolic systems where waves can propagate in both directions. First, we define a generalized numerical formula which can accommodate arbitrary wave speeds for scalar advection equation. Then to illustrate its application, we derive three, four, and five point generalization numerical schemes. Finally, according to the theory of linear systems, we extend the generalized schemes to linear hyperbolic systems in a straight forward manner.
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    Arbitrary-order numerical schemes for model parabolic equation
    (1992) Shi, Jian
    This report investigates the general theory and methodology of high order numerical schemes for one-dimensional model parabolic equation. The Universal Formula from which a 2-level explicit arbitrary-order numerical methods for diffusion equation can be derived is developed. Using the Universal Formula some high order numerical methods are constructed. Some important features of numerical methods are revealed through the construction of high order numerical methods and stability analysis. Subject to the limitation of diffusion number, d, being positive, only the method that satisfies positive stable region is relevant.
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    Fully discrete arbitrary-order schemes for a model hyperbolic conservation law
    (1993) Shi, Jian; Toro, E. F.
    We investigate the fully discrete methodology and establish a formula from which two-level explicit fully discrete arbitrary-order (both in space and time) conservative numerical schemes for a model hyperbolic conservation law can be derived. To illustrate this approach fully discrete second, third and fourth order numerical schemes are presented.
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    Fully discrete high resolution schemes for systems of conservation laws
    (Cranfield University, 1994-09) Shi, Jian; Toro, E. F.
    Effective and robust high resolution schemes are of vital importance for simulation of viscous and inviscid flows. Since second-order high resolution schemes in practice are inadquate for many applications, large efforts have been put towards developing higher- order accurate schemes in the past. Although some progress has been made, the efforts were frustrated by the lack of effective and robust new schemes. Therefore this thesis is aimed at challenging this difficult but very important issue. Some new theories and methodologies were established during this research, which covers the linear stability analysis for high-order numerical schemes; the fully discrete techniques for model equations; the formulation of conservative high-order schemes and the high-order Total Variation Diminishing (TVD) schemes. According to these theories arbitrary-order high resolution schemes can be developed. To illustrate the methodologies second-, third-, fourth-, and 20th-order schemes are presented. These high resolution schemes were tested and validated by solving some popular test problems for one and two dimensional Euler and incompressible Navier-Stokes equations. The efficiency and robustness are the features of these high-order schemes.
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    Fully discrete high-order schemes for hyperbolic conservation law.
    (1993) Shi, Jian
    In this paper we investigate fully discrete high-order accurate solutions for system of hyperbolic conservation laws. Second-, third- and fourth-order high resolution schemes are presented. Performance of the methods is assessed by solving test problems for time-dependent Euler equations of Gas Dynamics in one and two space dimensions. We use exact solutions and experimental data to validate the results.
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    Fully discrete high-order TVD schemes for a scalar hyperbolic conservation law
    (1993) Shi, Jian; Toro, E. F.
    In this paper we investigate fully discrete high-order TVD schemes for a scalar hyper- bolic conservation law using flux limiters . Formulae which define Courant number dependent TVD regions for second and third-order TVD schemes are established. A semi-empirical TVD procedure for an m-th order scheme (m ≥ 4) are proposed and tested.
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    A simplified Von Neumann method for linear stability analysis
    (1993) Shi, Jian
    This paper investigates the issue of linear stability analysis for two and three level explicit and implicit one dimensional finite different numerical schemes. A new approach which simplifies the Von Neumann method is presented. It has been proved that the new technique is efficient and effective for linear stability study. This is especially true for high-order and complicated numerical schemes.