The Space-Time Conservation Element and Solution Element (CESE) Method

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The space-time conservation element and solution element (CESE) method is a novel numerical method developed by Sin-Chung Chang for time-accurate, high-fidelity solution of hyperbolic partial different equations or conservation laws:

\[\frac{\partial\mathbf{u}}{\partial t} + \sum_{\iota=1}^3 \frac{\partial\mathbf{f}^{(\iota)}(\mathbf{u})}{\partial x_{\iota}} = 0 \quad \Rightarrow \quad \frac{\partial\mathbf{u}}{\partial t} + \sum_{\iota=1}^3 \mathrm{A}^{(\iota)}(\mathbf{u}) \frac{\partial\mathbf{u}}{\partial x_{\iota}} = 0\]

Key features of the CESE method include:

  • Temporal coordinate is treated in the same way as spatial coordinates for calculating space-time fluxes.
  • The method solves the integral form of the equations.
  • The method uses staggered space-time mesh.
  • Unstructured meshes are intrinsically used in multi-dimensional space.
  • The method is mathematically simple. Time-marching algorithms are based on enforcing space-time flux conservation.
  • No Riemann solver is used.
  • The gradients of solution are also independent unknowns.
  • The Jacobian matrices in the first-order formulation fully characterize the equations to be solved.

Consequently, the CESE method provides the following merits:

  • Nonlinearity is easily accommodated. Shock-capturing is accurate and robust.
  • The explicit time-marching algorithms are straightforward and efficient in terms of operation count.
  • Complex geometry can be easily fit by using unstructured meshes with mixed shapes.
  • The method is ideal for parallel computing.
  • The method is generic to all conservation laws, i.e., changing solely the Jacobian matrices changes the equations to be solved.
  • The method provides simple and effective treatments for non-reflective boundary conditions.