**Space-Time Conservation Element-Solution Element (CESE) Method**

**Fundamental concepts:**

- Solve integral form of the conservation law
- Enforce local and global flux conservation over discretized space-time domain
- Staggered integration volumes (conservation element) and solution volume (solution element), which allows:
- No cell interface Riemann solution needed
- No ad-hoc 1-D approximation at cell interfaces

- Solve both conservative flow variables and their spatial derivatives as independent unknowns
- Genuine multi-dimensional formation-- no dimensional/directional splitting is needed
- Non-dissipative baseline "a" scheme
- Numerical dissipation added to model actual physics with mathematical justification
- Naturally built for unstructured mesh (triangles for 2-D and tetrahedrons for 3-D)

**Avaiable numerical schemes:**

- "a" scheme
- "c" scheme
- Courant-number insensitive scheme (CNIS)
- 3rd-order scheme (under development)
- 4th-order scheme (under development)

**Refs.:**

- Chang, S.C. and To, W.M. (1991). A New Numerical Framework for Solving Conservation Laws -- The Method of Space-Time Conservation Element and Solution Element, NASA TM 104495.
- Chang, S.C. (1995). The Method of Space-Time Conservation Element and Solution Element - A New Approach for Solving the Navier-Stokes and Euler Equations. Journal of Computational Physics, Vol. 119, pp. 295-324.
- Chang, S.C., Wang, X.Y. and Chow, C.Y., (1999). The Space-Time Conservation Element and Solution Element Method -- A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws. Journal of Computational Physics, Vol. 156, No. 1, pp. 89-136.
- Chang, C.L. (2007). Three-Dimensional Navier-Stokes Calculations Using the Modified Space-Time CESE Method, AIAA Paper 2007-5818.

## Numerical topology:

**Spatial Derivative calculations**

**Courant Number Insensitive Scheme (CNIS)**

Solution point based (SCNIS)S.-C. Chang, NASA GRC |
Edge-based (ECNIS)C.-L. Chang, NASA LaRC |
Adjustable ECNISC.-L. Chang, NASA LaRC |