One of the main problems in fusion research is the understanding of the dynamics governing the heat transport in a tokamak plasma. Because of unexpectedly large transport coefficients observed in experiments the transport is called "anomalous". This property is one of the main reasons why there is no energy produced by fusion yet. Mathematically, the anomalous heat transport problem is modelled by a non-standard heat equation, with a heat conductivity coefficient depending on the gradient of the solution in a piecewise differentiable way with a jump discontinuity. In order to detect precisely the region where the anomalous transport starts playing a role, we develop an explicit front tracking technique. The differential equation is split at the discontinuity points (front points) into sub-problems. We prove that each of the problems can be treated separately and that their solutions match continuously at the inner boundary. To find the position of the inner boundary (the front point), we solve an additional ordinary differential equation. Numerically, we use the finite element method for the space discretization. For the time discretization, we apply a newly developed algorithm. It is based on the trapezoidal rule, but uses the defect of the 4th order Lobatto III A scheme for optimization of the number of the Newton iterations, adaptation of the time step and estimation of the local and global errors. The numerical capabilities of the algorithm are demonstrated on several numerical examples. The anomalous transport problem is solved and the parameter space is explored. The high accuracy of the algorithm is shown comparing the numerical and analytical results at the stationary state. Moreover, the treatment of multiple front points, which correspond to turbulence regime in the heat transport, is performed.
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One of the main problems in fusion research is the understanding of the dynamics governing the heat transport in a tokamak plasma. Because of unexpectedly large transport coefficients observed in experiments the transport is called "anomalous". This property is one of the main reasons why there is no energy produced by fusion yet. Mathematically, the anomalous heat transport problem is modelled by a non-standard heat equation, with a heat conductivity coefficient depending on the gradient of the...
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