One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

Semplice, Matteo; Travaglia, Elena; Puppo, Gabriella

doi:10.1007/s42967-021-00151-4

Cited by 6 publications

(5 citation statements)

References 45 publications

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“…Initial conditions read as R + (0, x) = sin(πx) and R − (0, x) = cos(πx). Furthermore, parameters for the CWENO reconstructions are chosen as Z-nonlinear weights as described in [33].…”

Section: Proposition 2 the Semilinear Boundary Value Problemmentioning

confidence: 99%

“…The computational framework is consistent with existing theoretical results, but also allows to consider problems which are beyond the current state of research on analytical control rules. The proposed method is based on high-order CWENO reconstructions [10,11,25,33] that are high-order accurate in smooth regions, but can resolve discontinuities in an essentially nonoscillatory (ENO) fashion. CWENO reconstructions consist of weighted combinations of local reconstructions on different stencils.…”

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confidence: 99%

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Numerical boundary control for semilinear hyperbolic systems

Gerster¹,

Nagel²,

Sikstel³

et al. 2022

Preprint

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This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.Introduction. Systems of hyperbolic partial differential equations model fluid flow, chemotaxis and viscoplastic material dynamics [2,23,26,34]. Boundary stabilization of these problems has been studied intensively in the past years [2,5]. An underlying tool for the study of these problems are Lyapunov functions that yield upper bounds on the deviation from steady states in suitable norms. The virtue of this approach are control rules that do not require the solution of the whole system, but take only measurements at the boundaries into account. So-called dissipative boundary conditions [6][7][8][9] ensure exponential decay of a continuous Lyapunov function, which in turn guarantees that the solution converges exponentially fast to a desired steady state.More precisely, a general theory for the stabilization of linear conservation laws with respect to the L 2 -norm is available [2, Sec. 3]. For nonlinear systems, however, results are still partial. A problem is posed by the fact that Lyapunov's indirect method [24] does not necessarily hold for hyperbolic systems. Furthermore, solutions to systems of conservation laws exist in the classical sense only for a finite time due to formation of shocks [31]. To this end, stability results are typically stated in terms of the Sobolev H 2 -norm or in the C 1 -norm [6,7,19,21] and restrictive smoothness assumptions on the H 2 -norm of the initial data may be needed [2, Sec. 4]. Furthermore, most analytical results are based on the assumption that the influence of the source term is small or in intuitive terms, the considered balance laws are viewed as perturbations of conservation laws [8]. On the other

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“…Initial conditions read as R + (0, x) = sin(πx) and R − (0, x) = cos(πx). Furthermore, parameters for the CWENO reconstructions are chosen as Z-nonlinear weights as described in [33].…”

Section: Proposition 2 the Semilinear Boundary Value Problemmentioning

confidence: 99%

mentioning

confidence: 99%

Numerical boundary control for semilinear hyperbolic systems

Gerster¹,

Nagel²,

Sikstel³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…This process is less expensive than the fully 2D WENO procedure. We note that there are other variants of the WENO procedure that can be mentioned, such as CWENO (Central WENO approach), as described in [24,25]; the recently introduced CWENOZ scheme [26], where boundary conditions are introduced without using ghost cells; or WENO scheme with Unconditionally Optimal Accuracy, as reported in [27].…”

Section: Spatial Weno Reconstructionmentioning

confidence: 99%

Modeling and Numerical Simulation of the Thermal Interaction between Vegetation Cover and Soil

Hidalgo

Tello

2022

Mathematics

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In this work, we propose a mathematical model representing the thermal interaction between vegetation cover and the soil underneath it. This model consists of a one-dimensional reaction–diffusion equation describing the evolution of the temperature in the vegetation cover coupled with a two-dimensional reaction–diffusion equation to represent the evolution of the temperature in the soil. The thermal interaction between the vegetation cover and the soil is studied and the distribution of temperatures in the soil with depth is also obtained. The vegetation cover acts in this model as a dynamic and diffusive boundary condition for the soil. The developed model takes into account the latent heat of fusion, which appears when the transformation of ice into liquid water or vice versa occurs inside the soil. The numerical approach for the solution of the mathematical model conducted in this work is based on the finite volume method with Weighted Essentially Non-Oscillatory technique for spatial reconstruction and the third-order Runge–Kutta Total Variation Diminishing numerical scheme is used for time integration, which is very efficient to obtain the numerical solution of this type of model. Some numerical examples are solved to obtain the distribution of temperature both in the vegetation cover and the soil.

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“…Such extrapolation can be based either on standard polynomial extrapolation or on WENO extrapolation [20]. Suitable WENO extrapolation procedures for ILW have been discussed in, e.g., [34,38].…”

Section: Time Dependent Hyperbolic Equationsmentioning

confidence: 99%

Inverse Lax-Wendroff Boundary Treatment: A Survey

Shu¹

2022

CMR

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The inverse Lax-Wendroff (ILW) procedure is a numerical boundary treatment technique, which allows finite difference schemes and other schemes to achieve stability and high order accuracy when using cartesian meshes to solve boundary value problems defined on complex computational domain. In this short survey we summarize the main ingredients of the ILW procedure, discuss its applicability and stability properties, and provide possible directions of its future development.

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One- and Multi-dimensional CWENOZ Reconstructions for Implementing Boundary Conditions Without Ghost Cells

Cited by 6 publications

References 45 publications

Numerical boundary control for semilinear hyperbolic systems

Numerical boundary control for semilinear hyperbolic systems

Modeling and Numerical Simulation of the Thermal Interaction between Vegetation Cover and Soil

Inverse Lax-Wendroff Boundary Treatment: A Survey

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