The Rate of Convergence of Some Difference Schemes

Hedstrom, G. W.

doi:10.1137/0705031

Cited by 58 publications

(26 citation statements)

References 12 publications

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“…Upon further analysis, this is a byproduct of the fact that the exact solution contains a jump and if the same test is performed for initial conditions defining a smooth transition between states the expected order of convergence, O(∆x), will result at some sufficient resolution. Knowledge of this phenomenon is not new and analysis is presented for example in [2,15,16,18,[18][19][20]. This example simply serves as useful demonstration of this type of behavior.…”

Section: The Essential Difficultymentioning

confidence: 99%

“…In [15], Hedstrom presents an analysis of general two level difference schemes and shows how the simple requirement of stability necessitates convergence at order p/(p + 1). This pioneering analysis demonstrated the fundamental behavior but failed to give any quantitative description of solution character other than asymptotic convergence rates.…”

Section: Introductionmentioning

confidence: 99%

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On sub-linear convergence for linearly degenerate waves in capturing schemes

Banks

Aslam

Rider

2008

Journal of Computational Physics

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A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2 , yielding a convergence rate of 1/2. Self-similar heuristics for higher order discretizations produce a growth rate for the layer thickness of ∆t 1/(p+1) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.

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Section: The Essential Difficultymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On sub-linear convergence for linearly degenerate waves in capturing schemes

Banks

Aslam

Rider

2008

Journal of Computational Physics

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“…It is important that the theorem applies to difference schemes stable in L2 (and not necessarily L00) since a difference scheme for (1) is stable in L°°o nly under very restrictive conditions (see Hedstrom [3], Thomée [8]). …”

Section: Jrmentioning

confidence: 99%

“…•'R Then û and vn, where u and vn axe given by (1) and (2) respectively, satisfy (3) û(9,t) = exp(ti9)û0(9), vn(9) = a(h9)"û0(9).…”

Section: Jrmentioning

confidence: 99%

“…Hedstrom [3], Serdyukova [5], Setter [6], Strang [7], Thomée [8], and the book [2] by Brenner, Thomée and Wahlbin). In Hedstrom [3] and Brenner, Thomée and Wahlbin [2] it is shown that if A is stable in L2, accurate of order r and if u is sufficiently smooth (measured in an appropriate Besov space), then vn -» u in L°°. Convergence is optimal with respect to the rate of convergence and the smoothness required of u0.…”

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Simplified 𝐿^{∞} estimates for difference schemes for partial differential equations

Layton¹

1982

Proc. Amer. Math. Soc.

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Abstract.L°° estimates for one-step difference approximations to the Cauchy problem for du/dt = du/dx are proven by means of simple L2-techniques. It is shown that, provided the difference approximation is stable in L2 (and not necessarily L°°) and accurate of order r. the error in approximating smooth solutions is 0(hr). This has been proven by Hedstrom and Thomée using Fourier multipliers and Besov spaces. The present paper shows how convergence rates in U° can be recovered using simple techniques (such as the Fourier inversion formula). The methods of Hedstrom and Thomée give sharper results when the difference scheme diverges. The present paper exploits the fact that estimates between L00 and Ú are frequently easier to obtain than between L00 and L°°.

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On the rate of convergence for parabolic difference schemes, II

Widlund¹

1970

Comm Pure Appl Math

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The Rate of Convergence of Some Difference Schemes

Cited by 58 publications

References 12 publications

On sub-linear convergence for linearly degenerate waves in capturing schemes

On sub-linear convergence for linearly degenerate waves in capturing schemes

Simplified 𝐿^{∞} estimates for difference schemes for partial differential equations

On the rate of convergence for parabolic difference schemes, II

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