2019 CSDMS meeting-028: Difference between revisions
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|CSDMS meeting abstract=In this study, | |CSDMS meeting abstract=In this study, implicit and explicit spectral solutions are considered for solving the linear diffusion term of a simple 2D loosely coupled landscape evolution model. Spectral methods are powerful tools for solving elliptical partial differential equations and are widely used in other fields, though they have received comparatively little attention in landscape evolution modelling. In the LEM considered, the land surface elevation is altered by three processes: regional uplift, fluvial incision, and linear hillslope diffusion. In the simplest case, these processes act in an undifferentiated way across the entire landscape. While a recent algorithm has provided a powerful implicit solution to for the fluvial incision term, explicit formulations of diffusion remain standard. However, when the desired grid is large, an explicit method may be restricted by stability to a time step too small for the timescales of interest. To solve this problem implicitly, I transform the problem into the spectral domain, solve the 2D diffusion equation with a Crank-Nicholson method, and compare the results to explicit finite difference and explicit spectral methods. In its most simple formulation, the spectral methods require periodic boundary conditions in both dimensions. Resulting from these conditions, I show a whimsically tessellating solution where the landscape takes the form of a flat torus. | ||
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Revision as of 11:31, 4 May 2019
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Implicit-spectral solution for a simple landscape evolution model
In this study, implicit and explicit spectral solutions are considered for solving the linear diffusion term of a simple 2D loosely coupled landscape evolution model. Spectral methods are powerful tools for solving elliptical partial differential equations and are widely used in other fields, though they have received comparatively little attention in landscape evolution modelling. In the LEM considered, the land surface elevation is altered by three processes: regional uplift, fluvial incision, and linear hillslope diffusion. In the simplest case, these processes act in an undifferentiated way across the entire landscape. While a recent algorithm has provided a powerful implicit solution to for the fluvial incision term, explicit formulations of diffusion remain standard. However, when the desired grid is large, an explicit method may be restricted by stability to a time step too small for the timescales of interest. To solve this problem implicitly, I transform the problem into the spectral domain, solve the 2D diffusion equation with a Crank-Nicholson method, and compare the results to explicit finite difference and explicit spectral methods. In its most simple formulation, the spectral methods require periodic boundary conditions in both dimensions. Resulting from these conditions, I show a whimsically tessellating solution where the landscape takes the form of a flat torus.
