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In this paper, we present first-order accurate numerical methods for solutionof the heat equation with uncertain temperature-dependent thermal conductivity.Each algorithm yields a shared coefficient matrix for the ensemble setimproving computational efficiency. Both mixed and Robin-type boundaryconditions are treated. In contrast with alternative, related methodologies,stability and convergence are unconditional. In particular, we proveunconditional, energy stability and optimal-order error estimates. A battery ofnumerical tests are presented to illustrate both the theory and application ofthese algorithms.

Unconditionally energy stable and first-order accurate numerical schemes for the heat equation with uncertain temperature-dependent conductivity