The Method of Manufactured Solutions for Code Verification

Roache, Patrick J.

doi:10.1007/978-3-319-70766-2_12

Cited by 16 publications

(10 citation statements)

References 34 publications

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“…The proposed scheme is numerically verified for convergence and accuracy by adopting the method of manufactured solutions (Cox, 2018; Grier et al , 2015; Oberkampf and Trucano, 2008; Roache, 2019). We consider the following coupled system: for y ∈ (0,1), t > 0 find u ( y , t ) and c ( y , t ) such that: where: …”

Section: Numerical Resultsmentioning

confidence: 99%

Analysis and application of a convergent difference scheme to nonlinear transport in a Brinkman flow

Nwaigwe

2020

HFF

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Purpose The purpose of this paper is to formulate and analyse a convergent numerical scheme and apply it to investigate the coupled problem of fluid flow with heat and mass transfer in a porous channel with variable transport properties. Design/methodology/approach This paper derives the model by assuming a fully developed Brinkman flow with temperature-dependent viscosity and incorporating viscous dissipation, variable transport properties and nonlinear heat and mass sources. For the numerical formulation, the nonlinear sources are treated in semi-implicit manner, whereas the non-constant transport properties are treated by lagging in time leading to decoupled diagonally dominant systems. The consistency, stability and convergence results are derived. The method of manufactured solutions is adopted to numerically verify the theoretical results. The scheme is then applied to investigate the impact of relevant parameters, such as the viscosity parameter, on the flow. Findings Based on the numerical findings, the proposed scheme was found to be unconditionally stable and convergent with first- and second-order accuracy in time and space, respectively. Physical results showed that the flow parameters have influence on the flow fields, particularly, the flow is enhanced by increasing porosity and viscosity parameters and the concentration decreases with increasing diffusivity, whereas both the temperature and Nusselt number decrease with increasing thermal conductivity. Practical implications Numerically, the proposed numerical scheme can be applied without concerns on time steps size restrictions. Non-physical solutions cannot be computed. Physically, the flow can be increased by increasing the viscosity parameters. Pollutants with higher diffusivity will have their concentration decreased faster than those of lower diffusivity. The fluid temperature would decrease faster if its thermal conductivity is higher. Originality/value A fully coupled fluid flow with heat and mass transfer problem having nonlinear properties and nonlinear fractional sources and sink terms, presumably, has not been investigated in a general form as done in this study. The detailed numerical analysis of this particular scheme for the identified general model has also not been considered in the past, to the best of the author’s knowledge.

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Section: Numerical Resultsmentioning

confidence: 99%

Analysis and application of a convergent difference scheme to nonlinear transport in a Brinkman flow

Nwaigwe

2020

HFF

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“…But it turns out that with the right trick, they are easy to createa technique called the "Method of Manufactured Solutions" (MMS). Rather than describe this method in detail, let us refer to Roache (2019); Salari and Knupp (2000); Jelinek and Mahaffy (2007); NETL Multiphase Flow Science Team (2020). In the end, the method provides us with an exact solution against which we can check our numerical solution for closeness and convergence.…”

Section: Create Known Solutionsmentioning

confidence: 99%

“…Once we have satisfied ourselves that the solution at least looks reasonable, it is time to verify that it actually is. This is best done by using a known solution, either because we have a simple-enough test case for which the solution can be derived analytically, or using the Method of Manufactured Solutions (Roache, 2019;Salari and Knupp, 2000;Jelinek and Mahaffy, 2007; NETL Multiphase Flow Science Team, 2020). If we know the exact solution (which we will denote by = ( ) or = ( , ) though concrete applications may of course use different symbols), we can compute the error in the numerical approximation ℎ through a norm such as the 2 norm,…”

Section: (D) If the Solution Looks Or Appears Wrong Let Usmentioning

confidence: 99%

I'm stuck! How to efficiently debug computational solid mechanics models so you can enjoy the beauty of simulations

Comellas¹,

Pelteret²,

Bangerth³

2022

Preprint

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A substantial fraction of the time that computational modellers dedicate to developing their models is actually spent trouble-shooting and debugging their code. However, how this process unfolds is seldom spoken about, maybe because it is hard to articulate as it relies mostly on the mental catalogues we have built with the experience of past failures. To help newcomers to the field of material modelling, here we attempt to fill this gap and provide a perspective on how to identify and fix mistakes in computational solid mechanics models.To this aim, we describe the components that make up such a model and then identify possible sources of errors. In practice, finding mistakes is often better done by considering the symptoms of what is going wrong. As a consequence, we provide strategies to narrow down where in the model the problem may be, based on observation and a catalogue of frequent causes of observed errors. In a final section, we also discuss how one-time bug-free models can be kept bug-free in view of the fact that computational models are typically under continual development.We hope that this collection of approaches and suggestions serves as a "road map" to find and fix mistakes in computational models, and more importantly, keep the problems solved so that modellers can enjoy the beauty of material modelling and simulation.

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“…As an exact solution to the pseudo‐impulsive radiation problem is non‐trivial, we consider the convergence study of the Laplace solver in the scope of the Method of Manufactured Solutions (MMS), 50 where a true solution,

{\phi}_k^{\mathrm{MMS}}

, is assumed to the problem, from which analytical expressions can be derived for the boundary conditions and right‐hand side function, and hereby reforming the Laplace problem into a Poisson problem. The true solution should be infinitely smooth, for example, a combination of trigonometric functions, to complement the wave problem, and with this, we seek to verify that we are able to achieve: (1) spectral

P

‐convergence for fixed meshes by increasing the order of orthonormal basis functions,

\psi

, and (2) algebraic

h

‐convergence of order

P+1

for fixed orders of orthonormal basis functions with decreasing element size.…”

Section: Numerical Propertiesmentioning

confidence: 99%

A spectral element solution of the two‐dimensional linearized potential flow radiation problem

Visbech

Engsig-Karup

Bingham

2022

Numerical Methods in Fluids

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We present a scalable two‐dimensional Galerkin spectral element method solution to the linearized potential flow radiation problem for wave induced forcing of a floating offshore structure. The pseudo‐impulsive formulation of the problem is solved in the time domain using a Gaussian displacement signal tailored to the discrete resolution. The added mass and damping coefficients are then obtained via Fourier transformation. The spectral element method is used to discretize the spatial fluid domain, whereas the classical explicit 4‐stage fourth‐order Runge–Kutta scheme is employed for the temporal integration. Spectral convergence of the proposed model is established for both affine and curvilinear elements, and the computational effort is shown to scale with 𝒪false(Npfalse), with N$$ N $$ being the total number of grid points and pprefix≈1.12$$ p\approx 1.12 $$. The solver is used to compute the hydrodynamic coefficients for several floating bodies and compared against known public benchmark results. The results show excellent agreement, ultimately validating the solver and emphasizing the geometrical flexibility and high accuracy and efficiency of the proposed solution strategy. Lastly, an extensive investigation of nonresolved energy from the pseudo‐impulse is carried out to characterize the induced spurious oscillations of the free surface quantities leading to a robust strategy for tuning the pseudo‐impulsive motion to the spatial discretization.

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The Method of Manufactured Solutions for Code Verification

Cited by 16 publications

References 34 publications

Analysis and application of a convergent difference scheme to nonlinear transport in a Brinkman flow

Analysis and application of a convergent difference scheme to nonlinear transport in a Brinkman flow

I'm stuck! How to efficiently debug computational solid mechanics models so you can enjoy the beauty of simulations

A spectral element solution of the two‐dimensional linearized potential flow radiation problem

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