Zigavaupotic at 14:24, 23 March 2024

2024-03-23T14:24:02Z

Zigavaupotic: Created page with "One of the most commonly mentioned drawbacks of the RBF-FD method is its stability. Furthermore, the RBF-FD method, due to scattered nodes, sometimes produces system (differen..."

2024-03-23T14:19:29Z

Created page with "One of the most commonly mentioned drawbacks of the RBF-FD method is its stability. Furthermore, the RBF-FD method, due to scattered nodes, sometimes produces system (differen..."

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One of the most commonly mentioned drawbacks of the RBF-FD method is its stability. Furthermore, the RBF-FD method, due to scattered nodes, sometimes produces system (differential) matrices with spurious eigenvalues. Those are far more common than with for example FVM or FEM method. The instabilities arise fastly even with linear PDEs.

With meshless methods (especially RBF-FD) one usually stabilises such schemes by adding a higher-order Laplacian (hyperviscosity) operator on the right-hand side.

$$
+\gamma \Delta^\alpha u
$$

The final scheme is referred to as hyperviscosity scheme

@@ Line 1: / Line 1: @@
-One of the most commonly mentioned drawbacks of the RBF-FD method is its stability. Furthermore, the RBF-FD method, due to scattered nodes, sometimes produces system (differential) matrices with spurious eigenvalues. Those are far more common than with for example FVM or FEM method. The instabilities arise fastly even with linear PDEs.
+One of the most common drawbacks of the RBF-FD method is its stability. Furthermore, the RBF-FD method, due to scattered nodes, sometimes produces system (differential) matrices with spurious eigenvalues. Those are far more common than with for example FVM or FEM method.
-With meshless methods (especially RBF-FD) one usually stabilises such schemes by adding a higher-order Laplacian (hyperviscosity) operator on the right-hand side.
+For such unstable systems, one usually considers numerical diffusion in the form of hyperviscosity. The hyperviscosity scheme introduces higher-order Laplacian operator (i.e. hyperviscosity operator) to the right hand side of the PDE
 $$
-  +\gamma \Delta^\alpha u
+  +\gamma \Delta^\alpha u,
 $$
-The final scheme is referred to as hyperviscosity scheme
+where $\gamma$ is a hyperviscosity constant and $\alpha$ is the order of hyperviscosity.

Hyperviscosity - Revision history

Zigavaupotic at 14:24, 23 March 2024

Zigavaupotic: Created page with "One of the most commonly mentioned drawbacks of the RBF-FD method is its stability. Furthermore, the RBF-FD method, due to scattered nodes, sometimes produces system (differen..."