The space-time Conservation Element and Solution Element (CESE) Method¶

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.

For more information, please consult the website of the space-time CE/SE working group.