| Summary: | In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mi>t</mi></msub><mo>+</mo><mi>α</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>x</mi></msub><mo>+</mo><mi>β</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>γ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mrow><mo>(</mo><msub><mi>p</mi><mn>3</mn></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>x</mi><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>η</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mrow><mo>(</mo><msub><mi>p</mi><mn>4</mn></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>x</mi><mi>x</mi><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model.
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