8. Stability, Accuracy and Error Analysis

Introduction and Context

A computational fluid dynamics (CFD) solution is not automatically reliable because a solver has produced a converged field of pressure, velocity, or temperature. Numerical results are approximations to the governing differential equations, and their usefulness depends on how faithfully the discrete equations represent the physical model, how robustly they are solved, and how sensitive the reported engineering quantities are to numerical choices.

Three questions must therefore be separated. First, are the continuous governing equations represented adequately by the discrete equations? Second, has the algebraic system produced by discretization been solved sufficiently accurately? Third, are the engineering conclusions—such as pressure drop, lift, drag, heat-transfer rate, or separation location—insensitive enough to numerical errors for the intended decision? These questions correspond broadly to discretization accuracy, iterative convergence, and verification of quantities of interest.

This chapter develops a systematic framework for analysing truncation, round-off, iterative, and discretization errors. It also examines stability, boundedness, conservation, mesh and time-step refinement, and the important distinction between convergence of equations, convergence of numerical solutions, and convergence of engineering outputs. The central message is that verification is not a single residual check. It is an evidence-based process of demonstrating that a computed result is numerically credible.

Learning Objectives

After studying this chapter, the reader should be able to:

1. distinguish modelling error from numerical error and identify the principal sources of numerical error in CFD;

2. derive and interpret truncation error for representative finite-difference and finite-volume approximations;

3. explain consistency, stability, convergence, boundedness, and conservation and relate these properties to one another;

4. assess residual reduction, iterative error, and algebraic convergence;

5. design mesh-refinement and time-step-refinement studies;

6. estimate observed order of accuracy and extrapolate a grid-independent quantity;

7. distinguish convergence of the discrete equations from convergence of a quantity of engineering interest;

8. identify common failure modes in verification and recommend professional best practices.

The Error Landscape in CFD

The exact physical problem consists of a mathematical model, boundary and initial conditions, and material properties. Its exact solution is rarely available for a realistic geometry. CFD replaces the continuous problem by a finite set of algebraic equations and then solves those equations approximately. Errors can arise at every stage.

Modelling error results when the mathematical description does not represent the physical system adequately. Examples include using a steady model for an inherently unsteady flow, neglecting compressibility, selecting an inappropriate turbulence model, or imposing unrealistic boundary conditions. Modelling error is not removed by refining the mesh.

Discretization error is the difference between the exact solution of the continuous mathematical equations and the exact solution of the discrete equations on a specified mesh and time step. It is caused by approximating derivatives, fluxes, integrals, geometry, and boundary conditions.

Iterative error occurs because the discrete algebraic equations are usually solved through an iterative procedure and terminated before the exact discrete solution is reached. It depends on solver tolerances, relaxation, convergence criteria, and the conditioning of the algebraic system.

Round-off error is introduced by finite-precision arithmetic. It is generally small in ordinary calculations, but can become important when subtracting nearly equal numbers, using very fine meshes, performing poorly scaled calculations, or running long unsteady simulations.

A useful conceptual decomposition is

\[

E_{total}=E_{model}+E_{disc}+E_{iter}+E_{round},

\]

although these terms do not always combine linearly and may interact. The expression is therefore a diagnostic framework rather than an exact error identity.

Truncation Error and Order of Accuracy

Truncation error is the local residual obtained when the exact solution of the differential equation is substituted into the discrete equation. Consider the one-dimensional second derivative. Taylor expansion about node \(i\) gives

\[

\phi_{i+1}=\phi_i+\Delta x\phi'_i+\frac{\Delta x^2}{2}\phi''_i+\frac{\Delta x^3}{6}\phi'''_i+O(\Delta x^4),

\]

with a corresponding expression for \(\phi_{i-1}\). Adding the two expansions gives

\[

\frac{\phi_{i+1}-2\phi_i+\phi_{i-1}}{\Delta x^2}

=\phi''_i+\frac{\Delta x^2}{12}\phi^{(4)}_i+O(\Delta x^4).

\]

The centred approximation is therefore second-order accurate in space, with a leading truncation error proportional to \(\Delta x^2\). A first-order backward approximation to a first derivative, by contrast, has a leading error proportional to \(\Delta x\).

The notation \(O(h^p)\) means that the error decreases in proportion to the \(p\)th power of a characteristic mesh size \(h\), as \(h\) tends to zero. If the mesh is uniformly refined by a factor of two, a second-order error should decrease by approximately a factor of four, provided the solution is smooth and the asymptotic refinement range has been reached.

Formal order is not identical to practical accuracy. A second-order scheme may have a larger error than a first-order scheme on a particular mesh if its error coefficient is large. Moreover, discontinuities, sharp layers, skewed cells, non-smooth geometry, limiters, and boundary treatments may reduce the observed order. In multiphysics computations, the overall order is often controlled by the least accurate component.

Consistency, Stability and Convergence

A discretization is consistent if its truncation error tends to zero as the mesh spacing and time step tend to zero. Consistency means that the discrete equations approach the intended differential equations in the refinement limit.

A numerical method is stable if errors introduced during computation do not grow without bound. These errors may arise from initial data, boundary data, round-off, truncation, or incomplete iteration. Stability is a property of the numerical method and its operating conditions; it is not the same as physical stability of the flow.

For linear well-posed initial-value problems, the Lax equivalence principle states, under appropriate assumptions, that consistency and stability are equivalent to convergence. Convergence means that the numerical solution approaches the exact solution of the continuous problem as the discretization is refined. In nonlinear CFD, complex boundaries, adaptive meshes, turbulence models, and iterative nonlinear solvers, the theorem is not a universal practical test, but its logic remains fundamental: a method must represent the equations consistently and control error growth.

For transient explicit schemes, stability often imposes a restriction on the time step. For the one-dimensional advection equation, a representative Courant number is

\[

C=\frac{u\Delta t}{\Delta x}.

\]

A particular discretization may be stable only when \(C\) lies within a specified range. The precise limit depends on the scheme and equation. A stable time step does not guarantee accuracy: a calculation can be stable but excessively diffusive if the time step is too large or if the spatial scheme is too dissipative.

Implicit methods are often less restrictive from a linear stability perspective, but they are not free from accuracy limitations, nonlinear convergence difficulties, or temporal discretization error. Increasing \(\Delta t\) may produce a numerically stable solution that fails to resolve vortex shedding, acoustic waves, thermal transients, or moving interfaces.

Boundedness, Monotonicity and Numerical Diffusion

A bounded numerical solution should not create unphysical new maxima or minima when the continuous problem and data do not support them. For a scalar transport equation, boundedness is closely related to monotonicity and positivity. These properties are especially important for concentration, density, turbulent kinetic energy, and species mass fractions.

Central differencing is formally second-order in smooth regions but may generate oscillations when convection dominates diffusion. First-order upwind differencing is generally more robust and bounded, but introduces numerical diffusion. Higher-resolution schemes attempt to combine accuracy in smooth regions with limiting near steep gradients. Flux limiters and total-variation-diminishing approaches modify the high-order flux to prevent spurious oscillations.

Numerical diffusion is not physical diffusion. It spreads gradients because of the discretization, mesh alignment, or time integration. It can delay separation, weaken vortices, smear shocks, and alter scalar mixing. Conversely, insufficient dissipation may produce nonphysical oscillations. Scheme selection is therefore a trade-off between boundedness, accuracy, conservation, and computational cost.

Conservation and the Finite-Volume Perspective

Conservation requires that the amount leaving one control volume through a shared face enters the neighbouring control volume with equal magnitude and opposite sign. In a finite-volume method, the discrete balance for a conserved variable \(\phi\) may be written as

\[

\frac{\rho_P V_P(\phi_P^{n+1}-\phi_P^n)}{\Delta t}

+\sum_f F_f\phi_f

=\sum_f D_f+S_PV_P,

\]

where the terms represent accumulation, convective flux, diffusive flux, and sources. If face fluxes are shared consistently, internal fluxes cancel when all control volumes are summed. The remaining imbalance is associated with boundaries, sources, and temporal accumulation.

Global conservation is necessary but not sufficient for accuracy. A highly diffusive scheme may conserve mass exactly while predicting an incorrect velocity profile. Local conservation is also important: a small global imbalance can conceal substantial local errors that cancel in the total.

Pressure–velocity coupling adds another consideration. A solver may report small momentum residuals while the continuity equation remains poorly satisfied, particularly in segregated algorithms. Mass-flow imbalance, pressure-correction convergence, and boundary flux consistency should therefore be monitored in addition to equation residuals.

Residuals, Iterative Error and What “Converged” Means

After discretization, the CFD problem becomes a nonlinear algebraic system

\[

\mathbf{R}(\mathbf{U})=\mathbf{0},

\]

where \(\mathbf{U}\) contains the discrete unknowns and \(\mathbf{R}\) is the residual vector. An iterative solver generates a sequence \(\mathbf{U}^{(k)}\). The residual norm indicates how well the current iterate satisfies the algebraic equations, but it does not directly measure the error relative to the exact continuous solution.

Residuals should be interpreted carefully. Their magnitude depends on scaling, units, equation normalization, mesh size, and source-term magnitudes. A reduction of six orders of magnitude is meaningful only if the residual is properly defined and the monitored engineering quantities have also stabilized. Conversely, a residual plateau may reflect a physically unsteady flow rather than solver failure.

A calculation may be considered algebraically converged when:

• scaled residuals have decreased to appropriate tolerances;
• mass, momentum, energy, and species imbalances are acceptably small;
• monitored forces, fluxes, temperatures, or pressure differences have stabilized or exhibit a statistically stationary pattern;
• further iterations do not change the reported quantities materially.

For a genuinely unsteady flow, the correct objective is not a constant instantaneous solution. Instead, each time step must be solved sufficiently, and time histories or statistical averages must be shown to be independent of the remaining iterative error.

Mesh and Time-Step Refinement

The most direct practical test of discretization error is refinement. A mesh study changes spatial resolution while holding other modelling and numerical choices as constant as possible. A time-step study changes \(\Delta t\) while maintaining adequate spatial resolution and comparable convergence tolerances.

A credible mesh study normally includes at least three systematically related meshes: coarse, medium, and fine. The refinement ratio should be stated, and the geometry, boundary conditions, solver settings, and physical models should not be changed arbitrarily between cases. Refinement should occur in regions important to the quantity of interest, such as boundary layers, wakes, recirculation zones, shocks, and high-gradient thermal regions.

The quantity of interest (QoI) may be a surface-averaged temperature, pressure loss, drag coefficient, maximum wall temperature, mass-flow rate, or integral heat-transfer coefficient. It should be selected before the study, because a mesh adequate for pressure drop may be inadequate for local wall shear stress.

For solutions in the asymptotic range, an observed order can be estimated from three results \(Q_1,Q_2,Q_3\) on successively refined meshes with ratio \(r=h_2/h_1=h_3/h_2\):

\[

p=\frac{\ln\left|\frac{Q_3-Q_2}{Q_2-Q_1}\right|}{\ln r}.

\]

The indexing must be defined clearly; here, \(Q_1\) is the finest and \(Q_3\) the coarsest result. If the refinement ratio is not constant, a generalized relation or regression should be used. The formula is unreliable when differences are dominated by noise, when the solutions are not in the asymptotic range, or when the QoI changes non-monotonically.

Richardson extrapolation estimates the limiting value for a uniform refinement sequence. For a second-order quantity with fine-grid result \(Q_f\) and refinement ratio \(r\),

\[

Q_{ext}=Q_f+\frac{Q_f-Q_c}{r^p-1},

\]

where \(Q_c\) is the coarser result. The estimated fine-grid uncertainty may be expressed through a grid-convergence index (GCI), which incorporates a safety factor and the estimated order. GCI is useful for reporting, but it is not a substitute for engineering judgement or a properly designed refinement study.

Time-step refinement follows the same logic. For an unsteady quantity, compare sufficiently converged solutions using progressively smaller time steps. The time step should resolve the dominant physical frequencies and satisfy the accuracy requirements of the temporal integration scheme. In moving-mesh or fluid–structure problems, refinement may need to include mesh motion, coupling iteration, and structural time scales.

Convergence of Equations, Fields and Engineering Quantities

Three types of convergence are often confused.

Equation convergence means that the residuals of the discrete equations are small. It concerns the algebraic solve.

Solution convergence means that the discrete field changes negligibly with further iterations, time advancement toward a steady state, or mesh refinement. It concerns variables such as velocity and temperature throughout the domain.

QoI convergence means that a selected engineering output is insensitive to further numerical refinement or iteration. It concerns the decision-relevant result.

These forms of convergence can disagree. A drag coefficient may appear stable while local wall shear remains inaccurate because positive and negative errors cancel in the surface integral. A pressure field may change visibly while the total mass flow remains constant. A residual may be low while a badly resolved recirculation region produces an incorrect separation length.

The correct verification target depends on the purpose of the simulation. Design optimization may require accurate gradients of drag or heat transfer. Safety analysis may prioritize maximum temperature or stress-coupled pressure. System-level performance may require a reliable integral flow rate rather than pointwise velocity. Verification should therefore be QoI-driven, while still checking fundamental conservation and field plausibility.

A Practical Verification Workflow

A professional workflow begins by defining the physical model, numerical method, expected scales, and QoIs. Before running a production case, perform a simple benchmark or manufactured-solution test when possible. Manufactured solutions are especially valuable because a known analytical field is inserted into the governing equations with constructed source terms, allowing the implemented order of accuracy to be measured directly.

Next, establish a mesh that resolves the important physical structures. Examine cell quality, non-orthogonality, aspect ratio, wall-normal spacing, and alignment with dominant flow direction. Select discretization schemes that are sufficiently accurate but remain bounded for the anticipated flow regime.

Run the solution with controlled stopping criteria. Record residual histories, conservation balances, monitor histories, solver iterations, and relevant dimensionless parameters. Do not change relaxation factors, schemes, or physical models midway without documenting the change.

Then perform mesh and time-step refinement. Compare both local fields and QoIs. Report percentage changes, estimated order where meaningful, and the final numerical uncertainty. Finally, compare the numerical uncertainty with other uncertainties, including material properties, boundary conditions, turbulence modelling, and experimental measurement uncertainty.

Illustrative Applied Case: Pressure Drop in a Heated Channel

Background

Consider an illustrative channel-flow study intended to predict pressure drop and wall heat-transfer coefficient in a heated internal passage. The flow is incompressible, turbulent, and statistically steady. The engineering decisions depend on the pressure penalty and whether the wall remains below a specified thermal limit.

Situation and Problem

An initial simulation on a relatively coarse mesh reports a pressure-drop coefficient of 1.84 and an average heat-transfer coefficient of 312 W m\(^{-2}\) K\(^{-1}\). The residuals decrease by more than five orders of magnitude, and the solver declares convergence. However, the wall temperature near a bend exhibits a steep gradient, and the predicted recirculation zone is only two cells long. The question is whether the result is sufficiently verified.

Analysis

Three meshes are generated with approximate refinement ratio 1.5: coarse, medium, and fine. The same turbulence model, wall treatment, inlet condition, convergence tolerances, and thermal boundary condition are retained. The monitored quantities are pressure-drop coefficient, area-weighted wall temperature, average heat-transfer coefficient, and global mass imbalance.

The pressure-drop coefficient changes from 1.84 on the coarse mesh to 1.91 on the medium mesh and 1.94 on the fine mesh. The heat-transfer coefficient changes from 312 to 328 and then 332 W m\(^{-2}\) K\(^{-1}\). The wall temperature maximum changes more substantially because it is a local QoI located in the recirculation region. Residual histories alone would not have revealed this sensitivity.

A time-step study is also performed because the flow contains weak periodic oscillations. The time step is halved twice while maintaining several iterations per time step and comparing averages over the same number of flow-through times. The mean pressure drop changes little, but the amplitude of the temperature oscillation changes noticeably on the largest time step.

Approach and Solution

The fine mesh is selected for production because the pressure-drop and average heat-transfer results are approaching an asymptotic trend, while the coarse result is not. The local maximum temperature is reported with a larger numerical uncertainty than the integral quantities. The final report includes mesh dimensions, refinement ratios, residual criteria, mass imbalance, averaging interval, time-step sensitivity, and a statement that turbulence-model uncertainty has not been eliminated by mesh refinement.

Outcome

The study demonstrates that algebraic convergence of the first simulation did not establish mesh independence. It also shows that different QoIs converge at different rates. The pressure-drop prediction is relatively robust, whereas the peak temperature requires finer near-wall and recirculation-region resolution.

Learning Points

The case illustrates five principles: residual reduction is necessary but insufficient; QoIs must be specified in advance; local extrema are often more sensitive than integral quantities; spatial and temporal errors must be separated; and numerical uncertainty should be reported alongside modelling uncertainty.

Common Failure Modes and Limitations

A frequent mistake is to call a result “grid independent” after comparing only two meshes. Two results can be accidentally close, particularly when errors are non-monotonic. Another mistake is to refine the mesh while simultaneously changing turbulence models, wall functions, under-relaxation, or boundary conditions, making the source of change impossible to identify.

Excessively refined meshes can increase round-off sensitivity, memory requirements, and solver difficulty. Poor-quality cells may make a nominally fine mesh less accurate than a coarser, better-designed one. Refinement that is uniform everywhere may waste resources while leaving the critical boundary layer unresolved.

Residual targets should not be copied blindly between problems. A residual of a given numerical magnitude has different significance in a low-speed flow, a compressible shock problem, and a strongly reacting system. Similarly, a steady solver may converge to one of several physically possible states, or converge to a steady state that is not physically realized because the true solution is unsteady.

Verification also has limits. It cannot validate an incorrect physical model, guarantee experimental agreement, or prove that an unresolved flow feature is absent. Validation requires comparison with reliable experimental or analytical evidence. Verification asks whether the equations were solved correctly; validation asks whether the correct equations were solved for the physical purpose.

Professional, Governance and Reproducibility Considerations

Engineering decisions based on CFD should retain an auditable record of mesh generation, solver version, discretization schemes, convergence criteria, physical models, boundary conditions, hardware, and post-processing definitions. Automated scripts for mesh studies and extraction of QoIs reduce transcription errors and improve repeatability.

Reported results should include uncertainty and limitations rather than presenting excessive precision. For example, a drag coefficient reported to six decimal places is misleading if mesh and turbulence-model uncertainty changes the third decimal place. Safety-critical applications may require independent review, benchmark cases, documented acceptance criteria, and controlled software configuration.

Automated optimisation and machine-learning surrogates introduce additional risks. A surrogate trained on numerically inconsistent data can reproduce solver artefacts rather than physical trends. Automated refinement should therefore use conservation checks, quality limits, and independent verification cases rather than relying solely on an error indicator.

Emerging Directions

Modern CFD increasingly uses adaptive mesh refinement, adjoint-based error estimation, high-order discretization, local time stepping, and uncertainty quantification. Adjoint methods are particularly attractive when a small number of QoIs must be improved, because they identify regions where discretization errors most strongly influence a selected output.

High-order methods can deliver excellent accuracy for smooth solutions, but shocks, contact discontinuities, complex industrial geometries, and robust boundary treatments remain challenging. Hybrid CPU–GPU solvers and automated workflows make larger refinement studies practical, but faster computation does not remove the need for numerical verification. The likely direction is integrated verification in which residual monitoring, conservation, adaptive refinement, uncertainty estimates, and reproducibility metadata are built into the simulation workflow.

Best-Practice Checklist

A defensible CFD study should:

• separate modelling, discretization, iterative, and round-off errors;
• define QoIs before selecting the final mesh;
• monitor residuals, conservation balances, and physical quantities together;
• use at least three systematically refined meshes when estimating order or grid uncertainty;
• perform temporal refinement for unsteady or time-accurate calculations;
• check boundedness, positivity, and absence of nonphysical oscillations;
• document mesh quality, wall resolution, schemes, solver tolerances, and averaging intervals;
• distinguish verification from validation;
• report uncertainty with an appropriate number of significant figures; and
• preserve sufficient information for independent reproduction.

Chapter Summary

CFD results contain several interacting sources of error. Modelling error concerns the physical assumptions; discretization error arises from approximating the governing equations; iterative error results from incomplete solution of the algebraic system; and round-off error originates in finite-precision arithmetic. Truncation error determines formal order of accuracy, but observed accuracy is influenced by smoothness, mesh quality, boundary treatment, limiters, and solver implementation.

Consistency ensures that the discrete equations approach the intended differential equations, while stability controls error growth. Boundedness and conservation are essential practical properties, but neither alone guarantees accuracy. Residual reduction demonstrates algebraic progress, not necessarily mesh independence or physical correctness.

Mesh and time-step refinement studies should be designed around engineering quantities of interest. Observed order, Richardson extrapolation, and grid-convergence measures are useful only when the solutions lie in a suitable asymptotic range. Equation convergence, field convergence, and QoI convergence are distinct and should be checked separately. Verification establishes numerical credibility; validation addresses physical adequacy. A mature CFD practice reports both results and their uncertainty, limitations, and reproducibility evidence.

Key Terms

Accuracy: Closeness of a numerical result to the exact or reference solution.

Algebraic convergence: Reduction of the residual of the discrete equations through iterative solution.

Boundedness: The property that a numerical solution does not create unacceptable new extrema or nonphysical values.

Consistency: The property that truncation error tends to zero as discretization scales tend to zero.

Conservation: Exact or controlled balance of conserved quantities across control-volume faces and boundaries.

Convergence: Approach of a numerical solution or QoI toward a limiting or exact value under iteration or refinement.

Discretization error: Difference between the exact continuous solution and the exact solution of the discrete equations.

Grid-convergence index: An estimate of numerical uncertainty based on systematic mesh refinement and observed order.

Iterative error: Error caused by stopping an iterative algebraic solver before reaching the exact discrete solution.

Modelling error: Error caused by assumptions or simplifications in the physical and mathematical model.

Monotonicity: A property preventing the numerical method from generating spurious oscillations or new extrema.

Numerical diffusion: Artificial smearing of gradients caused by discretization rather than physical transport.

Observed order: The rate at which a computed QoI changes under mesh or time-step refinement.

Quantity of interest: An engineering output selected for assessment, such as drag, pressure drop, or heat-transfer rate.

Residual: The imbalance remaining when a current numerical approximation is substituted into the discrete equations.

Richardson extrapolation: Estimation of a limiting solution using results from systematically refined discretizations.

Round-off error: Error introduced by finite-precision representation and arithmetic.

Stability: The ability of a numerical method to prevent errors from growing uncontrollably.

Truncation error: The local error produced when an exact differential-equation solution is inserted into a discrete approximation.

Verification: Demonstration that the governing equations have been solved correctly at the numerical level.

Review Questions

Short Answer Questions

1. Distinguish modelling error from discretization error in a CFD calculation.

2. What does the notation \(O(h^p)\) signify in a discretization analysis?

3. Why does a small residual not necessarily imply a mesh-independent solution?

4. Define boundedness and explain why it matters in scalar transport calculations.

5. State the difference between verification and validation.

Long Answer Questions

1. Explain consistency, stability, and convergence, and discuss their relationship in numerical solution methods.

2. Describe how truncation error is obtained using Taylor-series analysis for a centred second-derivative approximation.

3. Develop a complete mesh-refinement procedure for assessing pressure drop and wall heat transfer in an internal-flow problem.

4. Explain the role of conservation, boundedness, and numerical diffusion in selecting a convection discretization scheme.

5. Discuss why equation convergence, field convergence, and QoI convergence may produce different conclusions.

Analytical / Application Questions

1. Three successively refined meshes produce QoI values of 2.40, 2.28, and 2.25, with a constant refinement ratio of 2. Estimate the observed order and comment on whether the sequence appears asymptotic.

2. A steady CFD run has residuals reduced by six orders of magnitude, but the predicted drag continues to drift during additional iterations. Identify possible causes and propose a diagnostic procedure.

3. In a transient simulation, halving the time step changes the mean pressure drop by 0.3% but changes the peak wall temperature by 8%. Explain how the result should be reported and what further checks are appropriate.

22. Multiple Choice Questions with Answers

Question 1

Which error is directly associated with replacing derivatives and fluxes by algebraic approximations?

A. Modelling error

B. Discretization error

C. Round-off error

D. Experimental error

Correct Answer: B. Discretization error

Question 2

For a second-order spatial method, reducing the characteristic mesh size by a factor of two should ideally reduce the leading discretization error by approximately what factor?

A. Two

B. Three

C. Four

D. Eight

Correct Answer: C. Four

Question 3

What does a CFD residual primarily measure?

A. The difference between the numerical and experimental solutions

B. The remaining imbalance in the discrete algebraic equations

C. The uncertainty caused by the turbulence model

D. The spatial resolution of the mesh

Correct Answer: B. The remaining imbalance in the discrete algebraic equations

Question 4

Which statement best describes numerical diffusion?

A. Physical molecular diffusion represented exactly by the solver

B. Artificial smearing of gradients caused by the numerical method

C. Random error caused only by computer hardware

D. A correction applied to enforce global mass conservation

Correct Answer: B. Artificial smearing of gradients caused by the numerical method

Question 5

A mesh-refinement study should primarily compare which of the following?

A. Only the final residual values

B. Only the number of cells

C. Specified quantities of interest and relevant field behaviour

D. Different turbulence models on unrelated meshes

Correct Answer: C. Specified quantities of interest and relevant field behaviour

Question 6

What is the main purpose of a time-step refinement study?

A. To determine whether temporal discretization affects the result

B. To eliminate modelling error

C. To increase the formal order of the spatial scheme

D. To guarantee global conservation

Correct Answer: A. To determine whether temporal discretization affects the result

Question 7

A globally conserved flow can still be inaccurate because:

A. Conservation automatically prevents all numerical diffusion

B. Local errors can cancel in the global balance and the field may still be poorly resolved

C. Conservation applies only to experimental data

D. A conserved solution cannot contain boundary layers

Correct Answer: B. Local errors can cancel in the global balance and the field may still be poorly resolved

Question 8

Which result most directly indicates algebraic convergence?

A. The mesh contains a large number of cells

B. The residuals and discrete equation imbalances have become acceptably small

C. The numerical result agrees with a different turbulence model

D. The geometry has been imported without warning messages

Correct Answer: B. The residuals and discrete equation imbalances have become acceptably small

Question 9

In a refinement sequence, non-monotonic changes in a quantity of interest may indicate that:

A. The result is necessarily exact

B. The solutions may not be in the asymptotic refinement range or may contain competing errors

C. The method has zero truncation error

D. Round-off error has been completely eliminated

Correct Answer: B. The solutions may not be in the asymptotic refinement range or may contain competing errors

Question 10

Which statement correctly distinguishes verification from validation?

A. Verification assesses whether the equations were solved correctly; validation assesses whether the model represents reality adequately

B. Verification compares only experimental data; validation checks only residuals

C. Verification removes turbulence-model uncertainty; validation removes round-off error

D. They are identical terms for mesh refinement

Correct Answer: A. Verification assesses whether the equations were solved correctly; validation assesses whether the model represents reality adequately