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Completely discrete schemes for 2D Sobolev equations with Burgers’ type nonlinearity

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Abstract

Inthis paper, we discuss two first-order completely discrete schemes based on Backward Euler and its linearized variant methods for the 2D Sobolev equations with Burgers’ type nonlinearity. First, we derive some a priori estimates for the semi-discrete scheme, then a priori bounds for the fully discrete scheme are obtained for the backward Euler approximation. Use of discrete Gronwall’s Lemma and Stolz-Cesaro’s classical result for sequences show that these estimates for the fully discrete scheme are valid uniformly in time. Moreover, an existence of a global attractor of a discrete dynamical system is derived. Further, optimal a priori error bounds are established, which may depend exponentially on time. It is shown that these error estimates are uniform in time under a uniqueness condition. Moreover, as the coefficient of dispersion μ in − μΔut tends to zero, both the semi-discrete and completely discrete Sobolev equations converge to the corresponding Burgers’ equation linearly with respect to μ. Finally, some numerical examples are established in support of our theoretical analysis.

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Authors and Affiliations

  1. Department of Mathematics, S’O’A (Deemed to be University), Bhubaneswar, 751030, India

    Soumyarani Mishra

  2. Center of Applied Mathematics and Computation, S’O’A (Deemed to be University), Bhubaneswar, 751030, India

    Ambit K. Pany

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  1. Soumyarani Mishra
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Mishra, S., Pany, A.K. Completely discrete schemes for 2D Sobolev equations with Burgers’ type nonlinearity. Numer Algor 90, 963–987 (2022). https://doi.org/10.1007/s11075-021-01218-2

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