2. Method of Manufactured Solutions (MMS)

Order of accuracy testing (or ‘order testing’) is considered a rigorous method for performing code verification. During order testing, the formal order of accuracy of a numerical scheme is compared to the observed order of accuracy. The observed order is the order at which the discretization error (which is the difference between the numerical solution to the discrete equations and the exact solution to the PDEs) decreases with systematic mesh refinement. However, the exact solution to a PDE is unknown for most practical problems. In this scenario, the Method of Manufactured Solutions (MMS) [23] can be used where a selected analytical function (called a ‘manufactured solution’) is forced to be the exact solution by modifying the PDE through additional source terms.

The MMS test cases presented in this chapter are summarized in Table 2.1. These test cases offer an increasing level of computational complexity to isolate any potential problems in the source code. Three explanatory test cases that employ various simplifying assumptions are provided to assist the reader in understanding the MMS. Additional test cases were selected based upon their ability to invoke various parts of the MFIX code, and present physically acceptable data. All cases are executed in serial mode unless explicitly noted.

Table 2.1 Summary of MMS tests by feature.

	EX01	EX02	EX03	01	02	03	04	05	06
Frequency†	X	X	X	M	M	M	M	M	M
Dimension	1D	1D	1D	2D	3D	3D	3D	3D	3D
Multiphase					\({\checkmark}\)	\({\checkmark}\)
Continuity				\({\checkmark}\)	\({\checkmark}\)				\({\checkmark}\)
Momentum	\({\checkmark}\)			\({\checkmark}\)	\({\checkmark}\)	\({\checkmark}\)	\({\checkmark}\)	\({\checkmark}\)	\({\checkmark}\)
Thermal Energy		\({\checkmark}\)	\({\checkmark}\)		\({\checkmark}\)	\({\checkmark}\)
Species Mass
Granular Energy					\({\checkmark}\)	\({\checkmark}\)
Turbulence
No-slip Wall BC							\({\checkmark}\)
Free-slip Wall BC								\({\checkmark}\)
Pressure Outflow									\({\checkmark}\)
Distributed Memory
Shared Memory

† C-Incorporated into the continuous integration server; M-Monthly; Q-Quarterly; D-Disabled; X-Manual

MMS Procedure and Examples

• 2.1. MMS Procedure Overview
• 2.2. MMS-EX01: One dimensional steady state Burger’s equation
  • 2.2.1. Description
  • 2.2.2. Setup
  • 2.2.3. Results
• 2.3. MMS-EX02: One dimensional steady state heat equation
  • 2.3.1. Description
  • 2.3.2. Setup
  • 2.3.3. Results
• 2.4. MMS-EX03: One dimensional transient heat equation
  • 2.4.1. Description
  • 2.4.2. Setup
  • 2.4.3. Analysis variation for mixed variable problems
  • 2.4.4. Results
• 2.5. MMS01: Single-phase, 2D, sinusoidal functions
  • 2.5.1. Description
  • 2.5.2. Setup
  • 2.5.3. Results
  • 2.5.4. Notes
• 2.6. MMS02: Two-phase, 3D, curl-based functions with constant volume fraction
  • 2.6.1. Description
  • 2.6.2. Setup
  • 2.6.3. Results
• 2.7. MMS03: Two-phase, 3D, curl-based functions with variable volume fraction
  • 2.7.1. Description
  • 2.7.2. Setup
  • 2.7.3. Results
• 2.8. MMS04: No-slip wall BC, single-phase, 3D, curl-based functions
  • 2.8.1. Description
  • 2.8.2. Setup
  • 2.8.3. Results
• 2.9. MMS05: Free-slip wall BC, single-phase, 3D, curl-based functions
  • 2.9.1. Description
  • 2.9.2. Setup
  • 2.9.3. Results
• 2.10. MMS06: Pressure outflow BC, single-phase, 3D, curl-based functions
  • 2.10.1. Description
  • 2.10.2. Setup
  • 2.10.3. Results