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This paper presents a Monotonic Unbounded Schemes Transformer (MUST) approach to bound/monotonize (remove undershoots and overshoots) unbounded spatial differencing schemes automatically, and naturally. Automatically means the approach (1) captures the critical cell Peclet number when an unbounded scheme starts to produce physically unrealistic solution automatically, and (2) removes the undershoots and overshoots as part of the formulation without requiring human interventions. Naturally implies, all the terms in the discretization equation of the unbounded spatial differencing scheme are retained.

The authors do not formulate new higher-order scheme. MUST transforms an unbounded higher-order scheme into a bounded higher-order scheme.

The solutions obtained with MUST are identical to those without MUST when the cell Peclet number is smaller than the critical cell Peclet number. For cell Peclet numbers larger than the critical cell Peclet numbers, MUST sets the nodal values to the limiter value which can be derived for the problem at-hand. The authors propose a way to derive the limiter value. The authors tested MUST on the central differencing scheme, the second-order upwind differencing scheme and the QUICK differencing scheme. In all cases tested, MUST is able to (1) capture the critical cell Peclet numbers; the exact locations when overshoots and undershoots occur, and (2) limit the nodal value to the value of the limiter values. These are achieved by retaining all the discretization terms of the respective differencing schemes naturally and accomplished automatically as part of the discretization process. The authors demonstrated MUST using one-dimensional problems. Results for a two-dimensional convection–diffusion problem are shown in Appendix to show generality of MUST.

The authors present an original approach to convert any unbounded scheme to bounded scheme while retaining all the terms in the original discretization equation.

This viewpoint engages with Jem Bendell’s deep adaptation framework which was developed as a response to the threat of collapse. Proponents of deep adaptation argue that societal collapse is either likely, inevitable or already underway. The deep adaptation framework is employed as a tool to contemplate the necessary adaptation of tourism development and planning in a context of polycrisis leading to collapse.

This is a conceptual viewpoint article that is built on deductive analysis of recent events, reports and scientific findings. It employs the deep adaptation framework to analyse possible alternative tourism futures in the face of the threat of collapse.

Bendell’s framework included four aspects of response to the recognition of the threat of collapse: resilience, relinquishment, restoration and reconciliation. In this work, the deep adaptation framework is employed to analyse what a deep adaptation approach to tourism might offer for efforts in securing optimal social and ecological outcomes. Findings highlight damaging activities that we should relinquish, more resilient approaches that communities could encourage and restorative practices such as rewilding and pluriversal economies as protective measures. This work recommends a precautionary approach to transform tourism education, research and practice in order to secure better tourism futures.

This work is novel in engaging with the threat of future collapse and in using the deep adaptation framework to consider alternative tourism futures.