MMS: Compressible Euler Equations

Theoretical Foundations of MMS for the Compressible Euler Equations

Conservative Form and What Gets Verified

The compressible Euler equations are a conservation-law system for density, momentum, and total energy (two-dimensional, with source terms):

$$ \frac{\partial \mathbf{Q}}{\partial t} + \nabla\cdot\mathbf{F}(\mathbf{Q}) = \mathbf{S}, \qquad \mathbf{Q} = \begin{pmatrix} \rho \\ \rho\mathbf{u} \\ E \end{pmatrix}, \quad E = \frac{p}{\gamma - 1} + \frac{1}{2}\rho|\mathbf{u}|^2 $$

In MMS you pick a manufactured solution in the primitive variables \( (\rho_m, \mathbf{u}_m, p_m) \), substitute it into the left-hand side of the conservative form, and read off the source terms \( \mathbf{S} = (s_\rho, \mathbf{s}_{\rho u}, s_E) \) for each equation. With no viscous terms the derivation is shorter than for Navier-Stokes, but the difficulties peculiar to a hyperbolic system — characteristic-based boundary conditions, positivity, and interaction with shock-capturing schemes — are where the real verification battle is fought.

Designing the Manufactured Solution — Positivity and Smoothness

Characteristic Theory and How Many Conditions to Impose

In a hyperbolic system the number of conditions you may impose at a boundary is set by the number of incoming characteristics. For two-dimensional Euler the characteristic speeds are \( u_n - c,\ u_n,\ u_n,\ u_n + c \) — four of them, with \( c \) the speed of sound — and the count changes with the flow state at the boundary.

MMS obeys the same rule: correct verification means supplying manufactured values for only as many conditions as you are allowed to impose. Fixing every variable with Dirichlet data on every boundary over-specifies the subsonic case, producing a non-physical error layer near the boundary that contaminates the observed order. This is the single most common failure in compressible MMS.

Deriving Source Terms and Verifying Order

Derive in the Conservative Form

Derive the source terms in the form the solver actually discretizes. A finite volume solver is solving the conservative form, so deriving and injecting source terms for a non-conservative form written in primitive variables is poor verification practice even though the two are equivalent at the continuous level. With SymPy or similar the workflow is: (1) define the manufactured solution in primitive variables, (2) build the conservative variables and fluxes symbolically, (3) differentiate \( \partial_t \mathbf{Q} + \nabla\cdot\mathbf{F} \) to obtain each equation's source term, (4) apply common subexpression elimination and generate code. The energy source term is the longest, because pressure work and the derivative of kinetic energy both enter it, so always run a random-point cross-check against numerical differentiation.

Measure with Limiters and Artificial Viscosity Disabled

Compressible solvers ship with limiters and artificial viscosity for shock capturing. These can activate even on a smooth solution and locally drop the order to first, obstructing verification of the scheme's intrinsic order. The discipline is two-stage:

1. Stage 1: confirm the nominal order with limiters off (or with a limiter that is inactive in smooth regions) — this verifies the discretization itself
2. Stage 2: record how much order survives under production settings (limiters on) — this characterizes the error behavior you actually ship with. Losing order here is by design; the point is to know how much you lose

Evaluate Error Variable by Variable

Compute error norms separately for \( \rho, \rho u, \rho v, E \) (or the primitive variables \( \rho, u, v, p \)) and make every variable converging at the nominal order the pass criterion. If only one variable is low-order, the defect narrows down to the boundary treatment or flux evaluation involving that variable — energy alone being low, for instance, typically points to insufficient order in the temperature or pressure extrapolation at boundaries. When time integration is involved, separate the spatial and temporal verification just as with NS-MMS.

Applying It in Practice

Structure of a Standard Verification Campaign

1. 2-D subsonic MMS — a smooth field around Mach 0.3-0.5; confirm the order with characteristic boundaries (3 conditions on inflow, 1 on outflow) implemented
2. 2-D supersonic MMS — around Mach 1.5-3; fully specified inflow and fully extrapolated outflow exercise the other boundary path
3. Extension to 3-D — a manufactured solution with different wavenumbers per direction catches dimension-specific bugs such as swapped indices
4. Re-measurement under production settings — record the effective order with limiters and artificial viscosity enabled

These four cover the main paths: flux evaluation, boundary treatment, and the move to multiple dimensions. If you want to verify walls (slip boundaries), decide at the outset whether to design the manufactured velocity field to be wall-parallel at the boundary, or to treat the wall as a source-term boundary that permits transpiration.

Using MASA and Known Exact Solutions

The MASA library contains verified manufactured solutions for the Euler equations (2-D and 3-D, subsonic designs), which are useful as a cross-check against your own derivation. The compressible world also has an exact solution in the advecting isentropic vortex, a standard benchmark that measures the order of the time-evolution scheme without any source-term injection. "Isentropic vortex to confirm the basics, then MMS to cover boundaries and every term" is the practical two-step structure of a code verification campaign.

Reporting Format

The report should state: (1) the manufactured solution formulas with the amplitude and Mach number design, (2) the quantities imposed for each boundary type, (3) the mesh sequence and refinement ratio, (4) per-variable error norm tables with observed orders, and (5) results with limiters both enabled and disabled. Item (2) is the key to reproducibility: a verification report that does not say what was imposed at a subsonic inflow cannot be reproduced by anyone else.

How to Do It in Each Solver

Solver Comparison

Checklist of "Differences in Definition" to Settle Before Implementing

• Definition of total energy — is it \( E \) (total energy per unit volume), \( e \) (internal energy), or \( h_0 \) (total enthalpy)? The form of the energy source term changes accordingly
• Specific heat ratio and gas constant — the \( \gamma, R \) used in the derivation must match the solver settings. Beware implicit assumption of the air defaults (1.4 / 287)
• Nondimensionalization — if the solver works internally in nondimensional variables, nondimensionalize the source terms with the same reference quantities before passing them in
• Sign convention for source terms — added on the right-hand side or moved to the left? Follow how the equations are written in the manual

Research Frontiers

Verification Competition Among High-Order Schemes (DG, FR)

Because discontinuous Galerkin (DG) and Flux Reconstruction schemes carry nominal orders of four or six, demonstrating order via MMS has become entrenched in their development culture. At the international workshops that compare high-order CFD methods, the standard protocol is a side-by-side comparison of "error vs. degrees of freedom" and "error vs. compute time" on common MMS and isentropic vortex problems. MMS has thus moved beyond in-house verification to become a shared language for comparing schemes.

Entropy-Stable Schemes and Structure Preservation

As entropy-stable schemes — those satisfying the entropy inequality at the discrete level — have matured, the verification checklist has expanded from "order" alone to preservation of structure: the entropy budget, kinetic energy conservation, and free-stream preservation. Confirming the order with MMS and then layering on a free-stream preservation test (does a uniform flow stay undisturbed?) and entropy budget measurements is the modern standard for compressible solver development.

Verification in the Presence of Shocks

A smooth MMS cannot verify shock-capturing capability. Verification with discontinuities is an active research area: piecewise manufactured solutions that satisfy the shock relations exactly, exact solutions for moving shocks, and comparison against shock-fitted meshes. The practical division of labor is "order in smooth regions from MMS, shock position and strength from Riemann-problem benchmarks." Since the order of a global norm across a shock drops to first theoretically, demanding high order there is an error in the verification design.

Troubleshooting

Symptoms, Causes, and Fixes

The Shortest Path to Isolating the Fault

Related: automatic derivation of MMS source terms, troubleshooting NS-equation MMS, troubleshooting MMS convergence order.