| Summary: | In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mi>n</mi></msub></semantics></math></inline-formula>) and two Caputo fractional derivatives (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>) in time into the parabolic two-temperature model of the diffusive type. We prove that the obtained sub-diffusion two-temperature model is well posed. The numerical scheme is obtained based on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mn>1</mn></mrow></semantics></math></inline-formula> approximation for the Caputo fractional derivatives and the second-order finite difference for the spatial derivatives. Using the discrete energy method, we prove the numerical scheme to be unconditionally stable and convergent with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>,</mo><mn>2</mn><mo>−</mo><mi>β</mi><mo>}</mo></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>,</mo><mspace width="3.33333pt"></mspace><mi>h</mi></mrow></semantics></math></inline-formula> are time and space steps, respectively. The accuracy and applicability of the present numerical scheme are tested in two examples. Results show that the numerical solutions are accurate, and the present model and its numerical scheme could be used as a tool by changing the values of the Knudsen number and fractional-order derivatives as well as the parameter in the boundary condition for analyzing the heat conduction in porous media, such as porous thin metal films exposed to ultrashort-pulsed lasers, where the energy transports in phonons and electrons may be ultraslow at different rates.
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