Compact schemes for anisotropic reaction–diffusion equations with adaptive time step

Abstract

SUMMARYMany problems in biology and engineering are governed by anisotropic reaction-diffusion equations with a very rapidly varying reaction term. These characteristics of the system imply the use of very fine meshes and small time steps in order to accurately capture the propagating wave avoiding the appearance of spurious oscillations in the wave front. This work develops a fourth-order compact scheme for anisotropic reaction-diffusion equations with stiff reactive terms. As mentioned, the scheme accounts for the anisotropy of the media and incorporates an adaptive time step for handling the stiff reactive term. The high-order scheme allows working with coarser meshes without compromising numerical accuracy rendering a more efficient numerical algorithm by reducing the total computation time and memory requirements. The order of convergence of the method has been demonstrated on an analytical solution with Neumann boundary conditions. The scheme has also been implemented for the solution of anisotropic electrophysiology problems. Anisotropic square samples of normal and ischemic cardiac tissue have been simulated by means of the monodomain model with the reactive term defined by Luo-Rudy II dynamics. The simulations proved the effectiveness of the method in handling anisotropic heterogeneous non-linear reaction-diffusion problems. Bidimensional tests also indicate that the fourth-order scheme requires meshes about 45% coarser than the standard second-order method in order to achieve the same accuracy of the results.

Three-dimensional cardiac computational modelling: methods, features and applications

“…The FEM volumetric mesh was built using regular hexahedral elements (element size (ds) of 0.4 mm) in order to decrease the number of degrees of freedom of the model, yielding a mesh composed by 3.2 million elements and 3.5 million vertices. The final application of this model was computational simulation of cardiac EP using a specific FEM solver called ELVIRA [ 148 ], with a time step (dt) of 20 μs, and different ionic models for myocardium (including transmural heterogeneity) [ 68 ] and CCS [ 87 ] at cellular level, and monodomian approach [ 114 ] at tissue level.

Figure 10 Elements and final application of a 3D bi-ventricular patient-specific computational model.

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GPU accelerated solver for nonlinear reaction–diffusion systems. Application to the electrophysiology problem

“…The speed-up of the GPU implementation is compared against a fortran implementation of the algorithm described in Section 2 using the Message Passing Interface (MPI) for parallel computations [34,35]. Domain decomposition for the parallel solution has been carried out using the METIS library [36], where as the linear system of equations (12) has been solved using the Conjugate Gradient (CG) method with an ILU preconditioner from the PS-BLAS library [37].…”

Multiscale Modeling of Myocardial Electrical Activity: From Cell to Organ

“…The use of these ventricular 3D models has shed light into the mechanisms by which reentrant arrhythmias are initiated during regional myocardial ischemia. As an example, a 3D model of the human ventricles has been recently used to study the appearance of figureof-eight reentry under regional acute ischemic conditions (Heidenreich et al, 2010a;Heidenreich et al, 2010b). Fig.…”