The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion

We introduce a new higher order scheme for computing a tangentially stabilized curve shortening flow with a driving force represented by an intrinsic partial differential equation for an evolving curve position vector. Our new scheme is a combination of the explicit forward Euler and the fully-implicit backward Euler schemes. At any discrete time step, the solution is found efficiently using a few semi-implicit iterations. Basic properties of the new scheme are proved in the paper and its precision is tested by comparing the results with known analytical solutions. For any choice of the time step, the new higher order scheme gives exact radius of evolving uniformly discretized circles in case of flow by curvature and in case of rotation by a constant tangential velocity. Such properties do not hold for other schemes solving flow by mean curvature like the classical explicit, semi-implicit or fully-implicit schemes. In general, the scheme is second order accurate, which is shown by comparing a numerically evolving encompassed area with know analytical expression. The behavior of the scheme is discussed on representative examples and its advantages with respect to the balance between CPU time and precision are shown.

Abstract.A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.

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We present a second order accurate finite volume method for level set equation describing the motion in normal direction with the speed depending on external properties and curvature. A convenient combination of a Crank-Nicolson type of the time discretization for diffusion term [1] and an Inflow Implicit and Outflow Explicit scheme [6] for advection term is used. Numerical experiments for an example with the exact solution derived in this paper and for examples motivated by modeling of fire propagation in forests are presented.

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