Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Jeong, Darae; Li, Yibao; Lee, Chaeyoung; Yang, Junxiang; Choi, Yongho; Kim, Junseok

doi:10.1155/2019/8152136

Cited by 6 publications

(9 citation statements)

References 50 publications

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“…They urge their readers to be cautious of the varying order of convergence, but do not perform any detailed examination of the underlying cause of this discrepancy. The analysis presented in our paper explains the order of the convergence results in Jeong et al [8].…”

Section: Introductionsupporting

confidence: 63%

“…Moreover, the error will be diffusive in nature. Figure 1 shows the numerical solution of the linear advection equation (84) on the domain [0, 1] with wave speed a = 1, periodic boundary conditions, initial condition u(x, 0) = u 0 (x) = sin(2πx), and spatial resolution ∆x = 1/2 8 . The exact solution is u(x, t) = u 0 sin(2π(x − t)).…”

Section: Refinement Only In Space or Only In Timementioning

confidence: 99%

“…, 2 12 . For refinement only in time, we let N cells = 2 8 for the linear advection equation and N cells = 2 7 for the non-linear one. But we reduce ∆t from ∆t largest = ηtime ∆x to ∆t largest /2 5 by factors of 1/2.…”

Section: Convergence Plotsmentioning

confidence: 99%

“…In contrast, we present a more general and straightforward approach that is applicable to any discretization. Jeong et al [8] perform convergence tests of three parabolic PDEs-the heat equation, the Allen-Cahn equation, and the Cahn-Hilliard equation-by refining the spatial and temporal steps separately and together. They urge their readers to be cautious of the varying order of convergence, but do not perform any detailed examination of the underlying cause of this discrepancy.…”

Section: Introductionmentioning

confidence: 99%

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On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

Bishnu¹,

Peterson²,

Quaife³

2021

Preprint

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In this work, we determine the full expression of the local truncation error of hyperbolic partial differential equations (PDEs) on a uniform mesh. At a time horizon, the global truncation error, which is one order of the time step less than its local counterpart, assumes the formwhere ∆x and ∆t denote the cell width and the time step size, and α and β represent the orders of the spatial and temporal discretizations. If we are employing a stable numerical scheme and the global solution error is of the same order of accuracy as the global truncation error, we make the following observations in the asymptotic regime, where the truncation error is dominated by the powers of ∆x and ∆t rather than their coefficients. Assuming that we reach the asymptotic regime before the machine precision error takes over, (a) the order of convergence of stable numerical solutions of hyperbolic PDEs at constant ratio of ∆t to ∆x is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. We have tested our theory against numerical methods employing Method of Lines and not against ones that treat space and time together, and we have not taken into consideration the reduction in the spatial and temporal orders of accuracy resulting from slope-limiting monotonicity-preserving strategies commonly applied to finite volume methods. Otherwise, our theory applies to any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method. If the PDE is reduced to an ordinary differential equation (ODE) by specifying the spatial gradients of the dependent variable and the coefficients and the source terms to be zero, then the standard local truncation error of the ODE is recovered. We perform the analysis with generic and specific hyperbolic PDEs using the symbolic algebra package SymPy, and conduct a number of numerical experiments to demonstrate our theoretical findings.

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Section: Introductionsupporting

confidence: 63%

Section: Refinement Only In Space or Only In Timementioning

confidence: 99%

Section: Convergence Plotsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

Bishnu¹,

Peterson²,

Quaife³

2021

Preprint

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“…e authors in [20] presented two benchmark problems for phase-field models of solute diffusion and phase separation. Recently, a verification method for the convergence rates of the numerical solutions for wellknown parabolic partial differential equations was proposed in [23].…”

Section: Introductionmentioning

confidence: 99%

A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

Lee

Wang

et al. 2021

Discrete Dynamics in Nature and Society

Self Cite

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We present a very simple benchmark problem for the numerical methods of the Cahn–Hilliard (CH) equation. For the benchmark problem, we consider a cosine function as the initial condition. The periodic sinusoidal profile satisfies both the homogeneous and periodic boundary conditions. The strength of the proposed problem is that it is simpler than the previous works. For the benchmark numerical solution of the CH equation, we use a fourth-order Runge–Kutta method (RK4) for the temporal integration and a centered finite difference scheme for the spatial differential operator. Using the proposed benchmark problem solution, we perform the convergence tests for an unconditionally gradient stable scheme via linear convex splitting proposed by Eyre and the Crank–Nicolson scheme. We obtain the expected convergence rates in time for the numerical schemes for the one-, two-, and three-dimensional CH equations.

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A study about the solution of convection–diffusion–reaction equation with Danckwerts boundary conditions by analytical, method of lines and Crank–Nicholson techniques

Albesa

García

Ferrando

et al. 2022

Math Methods in App Sciences

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We compare different solutions of the convection-diffusion-reaction problem with Danckwerts boundary conditions. Analytical solution is found, and method of lines and Crank-Nicholson method are described, applied, and compared for different values of Péclet and Damköhler numbers. The eigenvalues and eigenfunctions have been obtained for all the possible values of the dimensionless parameters. And the analytical expression of the concentration has been calculated with the optimum number of terms in the Fourier expansion.

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Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Cited by 6 publications

References 50 publications

On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

On the Spatial and Temporal Order of Convergence of Hyperbolic PDEs

A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

A study about the solution of convection–diffusion–reaction equation with Danckwerts boundary conditions by analytical, method of lines and Crank–Nicholson techniques

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