High order implicit and explicit Lie-group schemes for solving backward heat conduction problems

This paper proposes high order implicit and explicit Lie-group schemes for solving backward heat conduction problems (BHCPs). The solutions to BHCPs are generally unstable and highly dependent on input data. Small perturbations in the input data can cause large oscillations in the solution. To overcome this problem, the variables in the nonlinear and nonhomogeneous heat conduction equation are transformed to obtain a fictitious time coordinate system by introducing a fictitious time variable and viscosity-damping coefficient. The original government equation can be transformed into a new heat conduction equation of an evolution type. Because the conventional scheme combined with the group preserving scheme (GPS) in Minkowski space must satisfy the constraint of the cone structure, Lie group and Lie algebra at each fictitious time step, solutions highly dependent on ones cannot easily converge. We use an evolution-type equation to calculate in Euclidean space and develop high order implicit and explicit Lie-group schemes. More importantly, for this difficult BHCP, by using both schemes, the problem becomes a linear problem of a single parameter, without choosing the parameters, such as the viscosity-damping coefficient, fictitious time step and the fictitious terminal time. Additionally, we modify the high order explicit Lie-group scheme to reduce the number of iterations in the implicit Lie-group scheme. The accuracy and efficiency of both schemes are validated, even under noisy measurement data, by comparing the estimation results with previous literature.