What is the Method of Manufactured Solutions?

A way to verify code even for equations without known exact solutions: choose any solution first (the manufactured solution), substitute it into the equation, and add what's missing as a source term. Solving with that source, check how close results come to the manufactured solution as the grid is refined, and whether error falls at the theoretical order.

What is the MMS procedure?

(1) Choose a smooth manufactured solution that exercises every term; (2) derive the source and boundary conditions symbolically; (3) solve with the source on several grids; (4) compute error and observed order and compare with the scheme's formal order. In the example, the correct implementation showed order 2.00.

What errors does MMS catch?

Even small errors whose results look plausible appear as a drop in convergence order. In the example, evaluating the face coefficient from one side instead of averaging dropped the order from 2.00 to 1.01. At 40 cells the error was about 1% — too small to notice from a plot.

Method of Manufactured Solutions (MMS) — Overview and Procedure

Category: V&V – Method of Manufactured Solutions | Revised 2026-10-01

CAE visualization for mms overview theory - technical simulation diagram

Overview and Procedure of MMS (Method of Manufactured Solutions)

Theory: choose the solution first

Overview

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I want to verify my own thermal code, but there's no exact solution for conductivity that varies with temperature. What can I do?

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Use the Method of Manufactured Solutions. Flip the approach: decide the solution you want first. Substituting it into the equation leaves a mismatch between the two sides; add that as a source term, and your chosen solution becomes exactly the solution of the modified equation. Then solve with that source in your code and check, as the grid is refined, whether error falls at the scheme's theoretical rate. The key is judging by how error decreases, not by whether the solution looks right.

The example equation

$$ -\frac{d}{dx}\left(k(u)\frac{du}{dx}\right) = f(x), \qquad k(u) = 1 + u^2, \qquad u_{MMS} = \sin(\pi x) + x $$

A nonlinear diffusion equation whose conductivity depends on the solution. The manufactured solution is smooth and equals 0 and 1 at the ends.

Source term and convergence order

$$ f(x) = -\frac{d}{dx}\left(k(u_{MMS})\frac{du_{MMS}}{dx}\right), \qquad p_{obs} = \frac{\ln(e_{2h}/e_h)}{\ln 2} $$

Derive the source symbolically (e.g. with sympy) to avoid hand-algebra slips. The observed order follows from the error ratio when spacing is halved.

Coffee Break Trivia

The reversal: manufacture the solution

Problems with analytical solutions are few and rarely cover the nonlinear, multidimensional, complex-boundary equations real software handles. In the 1980s Steinberg and Roache showed how to choose the solution first and fit the equation with a source term, and in the 2000s Roache, Salari, Knupp and others set it out as a standard procedure for code verification. It is now widely used to verify fluid codes including turbulence models, as well as solid, thermal and electromagnetic software, and ASME verification standards treat it as a central code-verification method.

Worked example: catching a small bug

With finite differences, a correct implementation evaluating face conductivity at the average of neighbouring values was compared with a faulty one using only the left value (a one-character difference). Solved by Picard iteration; maximum nodal error:

Cells	Correct error	Order	Faulty error	Order
10	1.13×10⁻²	—	4.62×10⁻²	—
20	2.85×10⁻³	1.98	2.13×10⁻²	1.11
40	7.10×10⁻⁴	2.00	1.04×10⁻²	1.03
80	1.78×10⁻⁴	2.00	5.15×10⁻³	1.02
160	4.44×10⁻⁵	2.00	2.56×10⁻³	1.01

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Even the faulty version has only about 1% error and it shrinks with refinement. I'd never notice.

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That's MMS's strength. Judged by error size alone, the faulty version seems roughly right and improving with refinement — on a plot it would almost overlap the manufactured solution. But the rate differs clearly: second order for the correct version, first for the faulty one. Evaluating the face coefficient one-sidedly injected first-order error into a nominally second-order scheme. In real problems that difference persists as a few percent of error even on fine grids. MMS brings invisible errors to the surface as a number: the order.

Procedure and tips

Choose a manufactured solution that exercises every term (convection, diffusion, source, nonlinear coefficients); overly simple solutions hide errors.
Derive source and boundary conditions symbolically and export them to code automatically.
Solve on four or more grids and compute error norms and orders.
Vary boundary conditions, grid types and nonlinear-iteration settings, checking each.
Turn results into automated tests that check the order on every code change.

Coffee Break Trivia

“First order only at the boundary”

Verifying a fluid code with MMS, interior error fell at second order, yet the overall maximum error fell only at first. The error map showed large error only at points next to walls — the boundary condition was implemented to first order, approximating the wall gradient with a one-sided difference. Switching to a second-order one-sided difference made the whole solution second order. Looking at the error distribution, not just its size, pinpoints where errors live.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Judging by error size alone	Order drop missed	Always compute order
Overly simple solution	Terms inactive, errors hidden	Exercise all terms
Hand-derived source	Errors in the source itself	Automate symbolically
Insufficient iteration convergence	Erratic order	Tight tolerance
Judging from two grids	Asymptotic range unknown	Four or more levels

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I'd like to learn related topics.

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Try convergence order with the Richardson extrapolation tool. Related pages include MMS convergence rates, MMS source terms, 2D heat conduction MMS, Navier–Stokes MMS and code verification.

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