This paper proposes a new relaxation-based circuit simulation algorithm called Backward-traversing Waveform Relaxation (BTWR). BTWR employs a brand new strategy to simulate: it simulates by performing Depth-First Search in the signal flow graph of simulated circuits. Major advantage of BTWR is the ability to flexibly schedule subcircuits for calculating according to subcircuits' converging situations, which enables BTWR to act robustly as well as efficiently in dealing with various types of circuits. The simulation on demand (SOD) function is also constructed in BTWR. All proposed methods have been implemented and tested by simulating various circuits.