Low regularity integrators for semilinear parabolic equations with maximum bound principles

Doan, Cao-Kha; Hoang, Thi-Thao-Phuong; Ju, Lili; Schratz, Katharina

doi:10.1007/s10543-023-00946-2

Cited by 2 publications

(15 citation statements)

References 57 publications

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“…Moreover, we have the following result on the fully discrete solutions produced by the LRI1 schemes (cf. [5] Theorem 2).…”

Section: Optimal Error Estimatesmentioning

confidence: 96%

“…$$ Setting

s=\Delta t

, we deduce that

{\boldsymbol{u}}_h\left({t}_{m+1}\right)={e}^{\Delta t{\boldsymbol{A}}_h}{\boldsymbol{u}}_h\left({t}_m\right)+\int_0^{\Delta t}{e}^{\left(\Delta t-s\right){\boldsymbol{A}}_h}\boldsymbol{F}\left({\boldsymbol{u}}_h\left({t}_m+s\right)\right) ds.

By approximating

{\boldsymbol{u}}_h\left({t}_m+s\right)

in (4) with

{e}^{s{\boldsymbol{A}}_h}{\boldsymbol{u}}_h\left({t}_m\right)

and using one‐point quadrature rules to compute the resulting integral, we obtain the following two first‐order LRI schemes (the reader is referred to [5] for the details):

\begin{matrix} alignleft \\ align-1 & align-2 i) LRI1a scheme: u_{h}^{m + 1} = e^{Δ t A_{h}} (u_{h}^{m} + Δ t F ( \end{matrix}

…”

Section: Fully Discrete First‐order Low Regularity Integratorsmentioning

confidence: 99%

“…Note that periodic and homogeneous Dirichlet boundary conditions also can be imposed instead as discussed later. Similar to [5, 6],

f

is first assumed to satisfy two conditions as follows: (A1)

f\left(\beta \right)\le 0\le f\left(-\beta \right)

for some contant

\beta >0.

(A2)

f

is continuously differentiable and nonconstant on

\left[-\beta, \beta \right]

. …”

Section: Introductionmentioning

confidence: 99%

“…LRIs are obtained by iteratively applying Duhamel's principle and also belong to the family of exponential integrators. The method was studied in [5] to solve the Allen‐Cahn type equations in combination with the central finite difference approximation in space and the proposed first‐ and second‐order LRI schemes are shown to preserve MBP and be energy stable. With a fixed spatial mesh, optimal temporal error estimates were derived for all the proposed schemes, given that the solution of the space‐discrete problem is only assumed to be continuous in time.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we aim to remove such dependence on the spatial mesh size in the error estimates and carry out the fully discrete error analysis of the two first‐order LRI schemes proposed in [5] when the time step size and the spatial mesh size simultaneously go to zero . In particular, we prove that, under assumptions (A1), (A2), and (A3), the fully discrete solution converges to the exact solution with first order in time and second order in space.…”

Section: Introductionmentioning

confidence: 99%

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Fully discrete error analysis of first‐order low regularity integrators for the Allen‐Cahn equation

Doan

Hoang

2023

Numerical Methods Partial

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The Allen-Cahn equation satisfies the maximum bound principle, that is, its solution is uniformly bounded for all time by a positive constant under appropriate initial and/or boundary conditions. It has been shown recently that the time-discrete solutions produced by low regularity integrators (LRIs) are likewise bounded in the infinity norm; however, the corresponding fully discrete error analysis is still lacking. This work is concerned with convergence analysis of the fully discrete numerical solutions to the Allen-Cahn equation obtained based on two first-order LRIs in time and the central finite difference method in space. By utilizing some fundamental properties of the fully discrete system and the Duhamel's principle, we prove optimal error estimates of the numerical solutions in time and space while the exact solution is only assumed to be continuous in time. Numerical results are presented to confirm such error estimates and show that the solution obtained by the proposed LRI schemes is more accurate than the classical exponential time differencing (ETD) scheme of the same order.

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“…Moreover, we have the following result on the fully discrete solutions produced by the LRI1 schemes (cf. [5] Theorem 2).…”

Section: Optimal Error Estimatesmentioning

confidence: 96%

“…$$ Setting

s=\Delta t

, we deduce that

{\boldsymbol{u}}_h\left({t}_{m+1}\right)={e}^{\Delta t{\boldsymbol{A}}_h}{\boldsymbol{u}}_h\left({t}_m\right)+\int_0^{\Delta t}{e}^{\left(\Delta t-s\right){\boldsymbol{A}}_h}\boldsymbol{F}\left({\boldsymbol{u}}_h\left({t}_m+s\right)\right) ds.

By approximating

{\boldsymbol{u}}_h\left({t}_m+s\right)

in (4) with

{e}^{s{\boldsymbol{A}}_h}{\boldsymbol{u}}_h\left({t}_m\right)

and using one‐point quadrature rules to compute the resulting integral, we obtain the following two first‐order LRI schemes (the reader is referred to [5] for the details):